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\begin{document}


Elementos

\begin{geothm}
Given four points $A$, $B$, $C$ and $D$, points of intersections are formed: $L = AB \cap CD$, $K = AC \cap BC$, $F = KL \cap BD$ and $G = KL \cap AC$. Show that segment $KL$ is harmonically divided by the points $F$ and $G$, ie: $\frac{\overline{LK}}{\overline{KF}} = \frac{\overline{GL}}{\overline{GF}}$.
\end{geothm}

\begin{geothm}
Two lines, $AB$ and $CD$ intersect at a point $O$. If $AC \parallel DB$, show that: $\frac{\overline{OA}}{\overline{OB}} = \frac{\overline{OC}}{\overline{OD}}$
\end{geothm}

\begin{geothm}
Given triangle $ABC$, let $p$ be line through $A$ parallel with $BC$, and let $q$ be line through $C$ parallel with $AB$. Let $p$ and $q$ intersects at $D$. Show that $\overline{AB} = \overline{DC}$.
\end{geothm}

\begin{geothm}
Let $ABC$ be an equilateral triangle inscribed in a circle with center $O$, and let $P$ be any point on the circle. Then the Simson line of $P$ bisects the radius $OP$.
\end{geothm}

\begin{geothm}
Starting from five points $A$, $B$, $C$, $D$ and $E$ with $A$, $B$, $C$ collinear, new lines and points of intersection are formed. $ED$, $IG$, $LK$ and $JH$ are collinear.
\end{geothm}

\begin{geothm}
In a cyclic quadrilateral having perpendicular diagonals, the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.
\end{geothm}

\begin{geothm}
Triangle $ABC$ and point $O$ are given. Let $p$, $q$ and $r$ be lines through the point $O$ which are perpendicular with lines $OA$, $OB$ and $OC$ respectively. Let $D$, $E$, and $F$ be intersections of lines $p$, $q$ and $r$ with lines $BC$, $AC$ and $BC$ respectively. Show that $D$, $E$ and $F$ are collinear.
\end{geothm}

\begin{geothm}
Given points $O$, $A$, $B$, $C$ and $O_1$ new points are constructed as: $A_1 = O_1B \cap OC$, $B_1 = OA \cap O_1C$, $C_1 = OB \cap O_1A$, $I = BB_1 \cap AA_1$. Show that points $C$, $C_1$ and $I$ are collinear.
\end{geothm}

\begin{geothm}
Let $C$ and $F$ be any points on the respective sides $AE$ and $BD$ of a parallelogram $ABCD$. Let $M$ and $N$ denote the points of intersection of $CD$ and $FA$ and of $EF$ and $BC$. Let the line $MN$ meet $DA$ at $P$ and $EB$ at $Q$. Then $AP \cong QB$.
\end{geothm}

\begin{geothm}
Prove that the line joining the point of intersection of the extensions of the nonparallel sides of a trapezoid to the point of intersection of its diagonals bisects the base of the trapezoid.
\end{geothm}

\begin{geothm}
Show that triangle altitudes are concurrent.
\end{geothm}

\begin{geothm}
Let $l$ be a line passing through the vertex of $M$ of a parallelogram $MNPQ$ and intersecting the lines $NP$, $PQ$, $NQ$ in points $R$, $S$, $T$. Show that $1/MR + 1/MS = 1/MT$.
\end{geothm}

\begin{geothm}
Let $A_1$, $B_1$, $C_1$ be points on the sides $BC$, $CA$, $AB$ of a triangle $ABC$ such that $BA_1/BC = CB_1/CA_1 = AC_1/AB = 1/3$. Show that the area of the triangle determined by lines $AA_1$, $BB_1$ and $CC_1$ is one seventh of the area of triangle $ABC$.
\end{geothm}

\begin{geothm}
Show that median $CM$ divides triangle $ABC$ into two triangles of the same area.
\end{geothm}

\begin{geothm}
Let $M$, $N$, $P$ be points on the sides $AB$, $BC$ and $AC$ of a triangle $ABC$. Show that if $M_1$, $N_1$ and $P_1$ are points on sides $AC$, $BA$ and $BC$ of a triangle $ABC$ such that $MM_1 \parallel BC$, $NN_1 \parallel CA$ and $PP_1 \parallel AB$, then triangles $MNP$ and $M_1N_1P_1$ have equal areas.
\end{geothm}

\begin{geothm}
Let $A_1$, $B_1$, $C_1$, $D_1$ be points on the sides $CD$, $DA$, $AB$, $BC$ of a parallelogram $ABCD$ such that $CA_1/CD = DB_1/DA = AC_1/AB = BD_1/BC = 1/3$. Show that the area of the quadrilateral formed by the lines $AA_1$, $BB_1$, $CC_1$, $DD_1$ is one thirteenth of the area of parallelogram $ABCD$.
\end{geothm}

\begin{geothm}
Two doubly perspective triangles are in fact triply perspective.
\end{geothm}

\begin{geothm}
The diagonals of a parallelogram and those of its inscribed parallelogram are concurrent.
\end{geothm}

\begin{geothm}
Given four points $A$, $B$, $C$ and $D$, points of intersections are formed: $L = AB \cap CD$, $K = AC \cap BC$, $F = KL \cap BD$ and $G = KL \cap AC$. Show that segment $KL$ is harmonically divided by the points $F$ and $G$, ie: $\frac{\overline{LK}}{\overline{KF}} = \frac{\overline{GL}}{\overline{GF}}$.
\end{geothm}

\begin{geothm}
Let $D$ be a point on the circumscribed cirle $(O)$ of triangle $ABC$. From $D$, three perpendiculars are drawn to the three sides $BC$, $CA$ and $AB$ of triangle $ABC$. Let $E$, $F$ and $G$ be the three feet respectively. Show that $E$, $F$ and $G$ are collinear.
\end{geothm}

\begin{geothm}
The cross ratio of four points on a circles with respect to any points on the circle is constant.
\end{geothm}

\begin{geothm}
In triangle $ABC$, let $F$ the midpoint of the side $BC$, $D$ and $E$ the feet of the altitudes on $AB$ and $BC$, respectively. $FG$ is perpendicular to $DE$ at $G$. Show that $G$ is the midpoint of $DE$.
\end{geothm}

\begin{geothm}
Let $P$ and $Q$ be two points on side $BC$ and $AD$ of a parallelogram such that $PQ \parallel AB$; $M = AP \cap BQ$, $N = DP \cap QC$. Show that $MN \parallel AD$ and $MN = AD/2$.
\end{geothm}

\begin{geothm}
Show that the sum of distances of a point on the base of an isosceles triangle to its two sides is equal to the altitudes on the sides.
\end{geothm}

\begin{geothm}
Given quadrilateral $ABCD$, let $M_1$ and $M_2$ be midpoints of diagonals $AC$ and $BD$, and let $O$ be point on line $M_1M_2$. Show that the sum of areas of triangles $ABO$ and $CDO$ equals the half of the area of quadrilateral $ABCD$.
\end{geothm}

\begin{geothm}
Of the three lines joining the vertices of an equilateral triangle to a point on its circumcircle, one is equal to the sum of the other two.
\end{geothm}

\begin{geothm}
Three parallel lines drawn through the vertices of a triangle $ABC$ meet the respective opposite sides in the points $X$, $Y$, $Z$. Show that area ratio $P_{XYZ} : P_{ABC} = 2 : 1$.
\end{geothm}

\begin{geothm}
Let $PT$ and $PB$ be two tangents to a circle, $AB$ the diameter through $B$, and $TH$ the perpendicular from $T$ to $AB$. Then $AP$ bisects $TH$.
\end{geothm}

\begin{geothm}
If a quadrangle $ABCD$ has its opposite sides $AD$ and $BC$ (extended) meeting at $W$, while $X$ and $Y$ are the midpoints of the diagonals $AC$ and $BD$, then $(WXY) = 1/4(ABCD)$.
\end{geothm}

\begin{geothm}
If lines $A_1B_1$, $A_2B_2$ and $A_3B_3$ are concurrent, show that intersections $A_1A_2 \cap B_1B_2$, $A_1A_3 \cap B_1B_3$ and $A_2A_3 \cap B_2B_3$ are collinear.
\end{geothm}

\begin{geothm}
Given triangle $ABC$, let $A_1$, $B_1$ and $C_1$ endpoints of triangle altitudes. From each of points $A_1$, $B_1$ and $C_1$ feet are drawn upon triangle sides, those are: $A_2$, $A_3$, $B_2$, $B_3$, $C_2$ and $C_3$. Show that these six points lie on a circle.
\end{geothm}

\begin{geothm}
If quadrilateral $ABCD$ is inscribed in circle, show that angles $\angle ACB$ and $\angle ADB$ are equal.
\end{geothm}

\begin{geothm}
Let $H$ and $O$ be the orthocenter and circumcenter of triangle $ABC$. Show $\angle HAC = | \angle B - \angle C |$.
\end{geothm}

\begin{geothm}
Hexagon $ABCDEF$ is incsibed in a circle with center $O$. Show that three points at which pairs of opposite sides meet, lie on a straight line.
\end{geothm}

\begin{geothm}
Let $ABC$ be a triangle with $\angle B = 2\angle C$, D the foot of the altitude on $CB$ and $M$ the midpoint of $B$ and $C$. Show that $AB = 2DM$.
\end{geothm}

\begin{geothm}
Given a triangle $ABC$ and a point $X$, the sum of the distances of the line $XG$, where $G$ is the centroid of $ABC$, to the two vertices of the triangle situated on the same side of the line is equal to the distance of the line from the third vertex.
\end{geothm}

\begin{geothm}
Given a parallelogram $ABCD$, a point $N$, obtained by the intersection of a line parallel to $BD$ passing through $C$, and a line perpendicular to $BD$ passing through $A$, then the point $Q$, which is given by the intersection of $BN$ and $CD$, is the midpoint of $PC$, where $P$ is the intersection of $CD$ and $AN$,
\end{geothm}

\begin{geothm}
Assume $S_1$, $S_2$, $S_3$ are the points of contact of the incircle of $ABC$: $S_1$ on side $AC$, etc. Denote the intersection of $BC$ and $S_1S_3$ as $P$, that of $AC$ and $S_2S_3$ as $Q$, and let $R$ be the intersection of $AB$ and $S_1S_2$. Then points $P$, $Q$, and $R$ are collinear.
\end{geothm}

\begin{geothm}
Line $p$ intersects lines $AB$, $AC$ and $BC$ at points $F$, $E$ and $D$ respectively. Show that:
\end{geothm}

\begin{geothm}
The midpoints of the three diagonals of a complete quadrilateral are collinear. This line is called Gauss line.
\end{geothm}

\begin{geothm}
A tranversal curs the sides $BC$, $CA$, $AB$ of triangle $ABC$ in $D$, $E$, $F$. $P$, $Q$, $R$ are the midpoints of $EF$, $FD$, $DE$ and $AP$, $BQ$, $CR$ intersec $BC$, $CA$, $AB$ in $X$, $Y$, $Z$. Show that $X$, $Y$, $Z$ are collinear.
\end{geothm}

\begin{geothm}
In a hexagon $AC_1BA_1CB_1$, $BB_1$, $C_1A$, $A_1C$ are concurrent and $CC_1$, $A_1B$, $B_1A$ are concurrent. Prove that $AA_1$, $B_1C$, $C_1B$ are also concurrent.
\end{geothm}

\begin{geothm}
In a triangle $ABC$, let $p$ and $q$ be the radii of two circles through $A$, touching side $BC$ at $B$ and $C$, respectively. Then $pq = R^2$.
\end{geothm}

\begin{geothm}
$OH^2 = 9R^2 - a^2 - b^2 - c^2$.
\end{geothm}

\begin{geothm}
The sum of distances of a point inside an equilateral triangle or on one of its sides to the three triangle segments, equals the length of its altitude.
\end{geothm}

\begin{geothm}
Show that every triangle has incenter.
\end{geothm}

\begin{geothm}
The feet of the four perpendiculars dropped from a vertex of a triangle upon the four bisectors of the other two angles are collinear.
\end{geothm}

\begin{geothm}
Isosceles right triangle $ABC$ with right angle at point $C$ is given. Let $E$ be a point on side $AC$ of a triangle. Let $D$ be the intersection of a line passing through $E$ parallel with line $AB$ and a line. $BC$. Let $p$ and $q$ be lines perpendicular with line $BE$ passing through points $D$ and $C$, and let $K$ and $L$ be intersections of lines $p$ and $q$ with hypotenuse $AB$. Show that $L$ is a midpoint of a segment $KA$.
\end{geothm}

\begin{geothm}
$P_1$, $P_2$, $P_3$ and $P_4$ are four points on circle $k$ with a center $O$. $M$ is the intersection of $P_1P_3$ and $P_2P_4$. Through $M$ draw a line $l$ perpendicular to $OM$, meeting $P_2P_3$ at $X$ and $P_1P_4$ at $Y$. Show that $MX \cong MY$.
\end{geothm}

\begin{geothm}
Prove that the angle over circle diameter is right angle.
\end{geothm}

\begin{geothm}
Let a quadrilateral $ABCD$ be inscribed in a circle. Then the sum of the products of the two pairs of opposite sides equals the product of its two diagonals. In other words, $AD \cdot BC + AB \cdot CD = AC \cdot BD$.
\end{geothm}

\begin{geothm}
There is a point $P$ inside of triangle $ABC$. Lines $PA$, $PB$ and $PC$ meet the triangle at three points $D$, $E$ and $F$. Show that: $\frac{\overline{AF}}{\overline{FB}} \cdot \frac{\overline{BD}}{\overline{DC}} \cdot \frac{\overline{CE}}{\overline{EA}} = 1$.
\end{geothm}

\begin{geothm}
Given triangle $ABC$, let $p$ be line through $A$ parallel with $BC$, and let $q$ be line through $C$ parallel with $AB$. Let $p$ and $q$ intersects at $D$. Let $BD$ intersect $AC$ at $E$. Show that: $\overline{AE} = \overline{EC}$.
\end{geothm}

\begin{geothm}
Show that each triangle has circumscribed circle.
\end{geothm}

\begin{geothm}
If we draw line segments between the midpoints, in order, of any quadrilateral, we get a parallelogram which has half the area of the original quadrilateral.
\end{geothm}

\begin{geothm}
Quadrilateral $ABCD$ is given such that diagonals $AC$ and $BD$ intersect at right angle. Let $A_1$, $B_1$, $C_1$ and $D_1$ be midpoints of quadrilateral sides $AB$, $BC$, $CD$ and $DA$ respectively. Let $A_2$, $B_2$, $C_2$ and $D_2$ be feet from points $A_1$, $B_1$, $C_1$ and $D_1$ upon lines $CD$, $DA$, $AB$ and $BC$ respectively. Show that eight points $A_1$, $B_1$, $C_1$, $D_1$, $A_2$, $B_2$, $C_2$ and $D_2$ lie on a circle.
\end{geothm}

\begin{geothm}
A line passing through the intersection $O$ of the diagonals of parallelogram $ABCD$ meets the four sides at $E$, $F$, $G$, $H$. Show that $EF \cong GH$.
\end{geothm}

\begin{geothm}
Let $D$ be a point on the side $CB$ of a right triangle $ABC$ such that the circle ($O$) with diameter $CD$ touches the hypotenuses $AB$ at $E$. Let $F = AC \cap DE$. Show that $AF \cong AE$
\end{geothm}

\begin{geothm}
Three equal squares are placed side by side, as it is shown on the picture. Show that: $\angle D_1AD + \angle D_1BD = \angle D_1CD$.
\end{geothm}

\begin{geothm}
From the ends $D$ and $C$ of a diameter of circle $(O)$ perpendiculars are drawn to chort $AB$. Let $E$ and $F$ be the feet of the perpendiculars. Show that $OE \cong OF$.
\end{geothm}

\begin{geothm}
Through $P$ a tangent $PE$ and a secant $PAB$ of circle $(O)$ are drawn. The bisector of angle $APE$ meets $AE$ and $BE$ at $C$ and $D$. Show that $EC \cong ED$.
\end{geothm}

\begin{geothm}
Assume the incircle of $\triangle ABC$ touches the sides $BC$, $AC$ and $AB$ in points $D$, $E$ and $F$ respectively. Show that lines $AD$, $BE$ and $CF$ meet at the Gergonne point $G$ of the triangle.
\end{geothm}

\begin{geothm}
Assume the incircle of $\triangle ABC$ touches the sides $BC$, $AC$ and $AB$ in points $D$, $E$ and $F$ respectively. The lines $AD$, $BE$ and $CF$ meet at the Gergonne point $G$ of the triangle. $DEF$ is known as Gergonne triangle (and also contact triangle) of $\triangle ABC$. Suppose three lines are drawn through $G$ parallel to the sides of the Gergonne triangle. These meet the sides of $\triangle ABC$ in six points $P$, $Q$, $R$, $S$, $T$ and $U$. Show that six points are concyclic. Moreover, the circle they lie on is centered at the incenter.
\end{geothm}

\begin{geothm}
Let $ABCD$ be a square. $CG$ is parallel to the diagonal $BD$. Point $E$ is on $CG$ such that $BE \cong BD$. $F$ is the intersection of $BE$ and $DC$. Show that $DF \cong DE$.
\end{geothm}

\begin{geothm}
Let $M$ be the midpoint of the arc $AB$ of circle $(O)$, $D$ be the midpoint of $AB$. he perpendicular through $M$ is drawn to the tangent of the circle at $A$ meeting that tangent at $E$. Show $ME \cong MD$.
\end{geothm}

\begin{geothm}
Let $G$ be a point on the circle $(O)$ with diameter $BC$, $A$ be the midpoint of the arc $BG$. $AD \perp BC$. $E = AD \cap BG$ and $F = AC \cap BG$. Show that $AE \cong BE \cong EF$.
\end{geothm}

\begin{geothm}
The two tangents to the circumcircle of $ABC$ at $A$ and $C$ meet at $E$. The mediator of $BC$ meet $AB$ at $D$. Show that $DE \parallel BC$.
\end{geothm}

\begin{geothm}
The product of two sides of a triangle is equal to the altitude to the third side multiplied by the circumdiameter.
\end{geothm}

\begin{geothm}
The internal and external bisectors of an angle of a triangle pass through the ends of the circumdiameters which is a perpendicular to the side opposite the vertex considered.
\end{geothm}

\begin{geothm}
On the hypotenuse $AB$ of right triangle $ABC$ a square $ABFE$ is erected. Let $P$ be the intersection of the diagonals $AF$ and $BE$ of $ABFE$. Show that: $\angle A C P = \angle P C B$.
\end{geothm}

\begin{geothm}
A line parallel to the base of trapezoid $ABCD$ meet ifs two sides and two diagonals at $H$, $G$, $F$ and $E$. Show that: $EF \cong GH$.
\end{geothm}

\begin{geothm}
Let $D$ and $E$ be two points on two sides $AC$ and $BC$ of triangle $ABC$ such that $AD \cong BE$. $F = DE \cap AB$. Show that: $FD \cdot AC = EF \cdot BC$.
\end{geothm}

\begin{geothm}
Two parallel line $AE$, $BD$ through the vertices $A$, $B$ of the triangle $ABC$ meet a line through the vertex $C$ in the points $E$, $D$. If the parallel through $E$ to $BC$ meets $AB$ in $F$, show that $DF$ is parallel to $AC$.
\end{geothm}

\begin{geothm}
Let $C$ be the midpoint of the arc $AB$ of circle $(O)$. $D$ is a point on the circle. $E = AB \cap CD$. Show that: $CA^2 = CE \cdot CD$.
\end{geothm}

\begin{geothm}
Through point $F$ on the circle with diameter $AB$ a tangent to the circle is drawn meeting the two lines, perpendicular to $AB$ at $A$ and $B$, at $D$ and $E$. Show that: $OA^2 = DF \cdot EF$.
\end{geothm}

\begin{geothm}
From a point $A$ two lines are drawn tangent to circle $(O)$ at $B$ and $C$. From a point $P$ on the circle peprendiculars are drawn to $BC$, $AB$ and $AC$. Let $D$, $F$, $E$ be the feet. Show that: $PD^2 = PE \cdot PF$.
\end{geothm}

\begin{geothm}
Of $Q$, $R$ are the projections of a point $M$ of the internal bisector $AM$ of the angle $A$ of the triangle $ABC$ upon the sides $AC$, $AB$, show that the perpendicular $MP$ from $M$ upon $BC$ meets $QR$ in the point $N$ on the median $AA_1$ of $ABC$.
\end{geothm}

\begin{geothm}
If $M$, $N$ are points on the sides $AC$, $BC$ of a triangle $ABC$ and the lines $AN$, $BM$ intersect on the altitude $CD$, show that $CD$ is the bisector of the angle $\angle MDN$.
\end{geothm}

\begin{geothm}
Show that the internal (or external) bisector of an angle of a triangle is divided harmonically by the feet of the perpendiculars dropped upon it from the two other vertices of the triangle.
\end{geothm}

\begin{geothm}
The angle between the circumdiameter and the altitude issued from the same vertex of a triangle is bisected by the bisector of angle of the triangle at the same vertex considered.
\end{geothm}

\begin{geothm}
The area of a triangle is equal to the product of its three sides divided by the double circumdiameter of the triangle.
\end{geothm}

\begin{geothm}
Let $M$ be the midpoint of chord $AB$ of a circle with center $O$. On $OM$ as diameter draw another circle, and at any point $T$ of this circle draw a tangent to it meeting the outer circle in $E$. Prove that: $AE^2 + BE^2 = 4ET^2$.
\end{geothm}

\begin{geothm}
Line $p$ intersects lines $AB$, $BC$, $CD$ and $DA$ at points $A_1$, $B_1$, $C_1$ and $D_1$ respectively. Show that: $\frac{\overline{AA_1}}{\overline{A_1B}} \cdot \frac{\overline{BB_1}}{\overline{B_1C}} \cdot \frac{\overline{CC_1}}{\overline{C_1D}} \cdot \frac{\overline{DD_1}}{\overline{D_1A}} \cdot = 1$
\end{geothm}

\begin{geothm}
Let $ABC$ and $A_1B_1C_1$ be two lines such that $AB_1 \parallel BA_1$ and $BC_1 \parallel CB_1$. Show that $AC_1 \parallel CA_1$.
\end{geothm}

\begin{geothm}
Let $ABC$ and $A_1B_1C_1$ be two lines and $P = AB_1 \cap A_1B$, $Q = AC_1 \cap A_1C$, $S = BC_1 \cap B_1C$. Then $P$, $Q$ and $S$ are collinear.
\end{geothm}

\begin{geothm}
Show that the distance of a point on a median of triangle from the sides including the median are inversely proportional to these sides, ie: $NK : NJ = BC : AC$.
\end{geothm}

\begin{geothm}
If $G$ is a centroid of triangle $ABC$, show that for any given point $M$: $3 \cdot MG^2 + AG^2 + BG^2 + CG^2 = AM^2 + BM^2 + CM^2$.
\end{geothm}

\begin{geothm}
Two tritangent centers divide the bisector on which they are located, harmonically (ie, points $C$, $D$, $I$ and $I_c$ are harmonic).
\end{geothm}

\begin{geothm}
Show that in any given triangle the ratio between one triangle side and a diameter of circumscribed circle is equal to the sine of opposite angle, ie: $\frac{a}{2R} = \sin \alpha$
\end{geothm}

\begin{geothm}
$c^2 + b^2 - 2bc \cos \alpha = a^2$
\end{geothm}

\begin{geothm}
If $R$ and $r$ are half diameters of circumscribed and inscribed circles of a trougle and $d$ is a distance between them, show that: $d^2 = R(R - 2r)$.
\end{geothm}

\begin{geothm}
Show that the area of a right triangle is equal to the product of the two segments into which the hypothenuse is divided by its point of contact with incircle.
\end{geothm}

\begin{geothm}
The segment of the altitude extended between the orthocenter and the second point of intersection with the circumcircle is bisected by the corresponding side of the triangle.
\end{geothm}

\begin{geothm}
Show that the mediator of the bisector $CU$ of the triangle $ABC$, the perpendicular to $AB$ at $U$, and the circumdiameter of $ABC$ passing through $C$ are concurrent.
\end{geothm}

\begin{geothm}
Let $A_1$ and $B_1$ be midpoints of segments $AC$ and $BC$ in triangle $ABC$. Show that $A_1B_1$ is parallel to $AB$ and half of that segment.
\end{geothm}

\begin{geothm}
Cross ratio of four points $A$, $B$, $C$ and $D$ is same after central projection from point $O$ onto line $p$.
\end{geothm}

\begin{geothm}
Let $A$, $B$ and $C$ be three points on the line $p$, and let $A_1$, $B_1$ and $C_1$ be three points on the line $q$. Following intersections are created: $P = AB_1 \cap A_1B$, $Q = AC_1 \cap A_1C$ and $R = BC_1 \cap B_1C$. Show that points $P$, $Q$ and $R$ are collinear.
\end{geothm}

\begin{geothm}
Show that in any triangle, the orthocenter, the centroid and the center of circumscribed circle are collinear.
\end{geothm}

\begin{geothm}
Prove that in any triangle midpoints of each side, feet of each altitude and midpoints of the segments of each altitude from its vertex to the orthocenter lie on circle.
\end{geothm}

\begin{geothm}
Segment $AB$ is given. Let $m$ be median of $AB$ and let circle $k$ be circle with center $A$ and half-diameter $B$. If $C$ is the intersection of circle $k$ and median $m$, show that triangle $ABC$ is isosceles.
\end{geothm}



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