الهندسة – مبادرة نجب

It is recommendable to introduce the power of a point by solving the following exercises:

Exercise 1. Let $P$ be a point outside a circle $\omega$ and $PT$ its tangent. Let an arbitrary line $l$ through $P$ intersects $\omega$ at the points $X$ and $Y$ . Show that $PX \cdot PY = PT^2 = OP^2 - R^2,$ where $R$ is the radius of $\omega$ .
Exercise 2. Let $P$ be a point inside a circle $\omega$ . Let an arbitrary line $l$ through $P$ intersects $\omega$ at the points $X$ and $Y$ . Show that $PX \cdot PY = R^2 - OP^2,$ where $R$ is the radius of $\omega$ .

The key message to deliver is that the product $PX \cdot PY$ is constant, i.e., it does not depend on $l$ .

1. Power of a Point

Review the properties of cyclic quadrilaterals, tangent angles, and similar triangles.

Let $P$ be a point and $\omega$ a circle with center $O$ and radius $R$ .

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Definition 1. The quantity $p(P, \omega) = OP^2 - R^2$ is called the power of point $P$ with respect to the circle $\omega$ . If $p(P, \omega) > 0$ , then $P$ lies outside $\omega$ , and if $p(P, \omega) < 0$ , then $P$ lies inside $\omega$ . The locus of points with a given power of point $p$ is a circle with center $O$ and radius $\sqrt{R^2 + p}$ .

Theorem 1: Criterion for Concyclity. Let $ABCD$ be a convex quadrilateral. Let the lines $AB$ and $CD$ intersect at $P$ , and the diagonals $AC$ and $BD$ intersect at $R$ . Then the following are equivalent:

i) $ABCD$ is a cyclic quadrilateral
ii) $PA \cdot PB = PC \cdot PD$
iii) $RA \cdot RC = RB \cdot RD$

Exercise 3. Let $ABC$ be a triangle and $D$ a point on the line $BC$ outside of the segment $BC$ . Then $DA$ is tangent to the circumcircle of $\triangle ABC$ if and only if $DA^2 = DB \cdot DC$ .
Exercise 4. Let two given circles $\omega_1$ and $\omega_2$ meet externally at the points $A$ and $B$ . Let $MN$ be their common tangent with $M \in \omega_1$ and $N \in \omega_2$ . Prove that the line $AB$ passes through the midpoint of $MN$ .
Exercise 5. Let $O$ and $I$ be the circumcenter and the incenter of a triangle $ABC$ , respectively. Let $R$ and $r$ be the circumradius and the inradius of $\triangle ABC$ , respectively. Show that $OI^2 = R^2 - 2Rr$ and $R \ge 2r$ .
Exercise 6. Let $ABC$ be an acute triangle. The line through $B$ perpendicular to $AC$ intersects the circle with diameter $AC$ at points $P$ and $R$ , and the line through $C$ perpendicular to $AB$ intersects the circle with diameter $AB$ at points $Q$ and $S$ . Show that points $P, Q, R,$ and $S$ lie on the same circle.

2. Radical Axis

Review known loci of points in geometry.

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Proposition 2: Radical Axis. Let $\omega_1(O_1, R_1)$ and $\omega_2(O_2, R_2)$ be two circles such that $O_1 \not\equiv O_2$ . The locus of the points $P$ such that $p(P, \omega_1) = p(P, \omega_2)$ is a line perpendicular to the line $O_1 O_2$ .

Methodological Suggestions: If two circles intersect in two points, their radical axis is the line through the intersection points. If two circles are mutually tangent, their radical axis is the tangent at the common point.

Theorem 3: Radical Axis Theorem. Let $\omega_1, \omega_2,$ and $\omega_3$ be three circles with distinct centers. Let $l_1$ be the radical axis of $\omega_2$ and $\omega_3$ , $l_2$ the radical axis of $\omega_3$ and $\omega_1$ , and $l_3$ the radical axis of $\omega_1$ and $\omega_2$ . Then the lines $l_1, l_2,$ and $l_3$ are either concurrent or parallel. The intersection point of the radical axes of three circles is called the radical center of the three circles.

Methodological Suggestions: The radical axis theorem is a powerful tool to show that three lines are concurrent. Also, the notion of radical axis may serve for showing that three or more points are collinear by identifying the line through them as the radical axis of two circles.

Exercise 7. The tangent line of the circumscribed circle of $\triangle ABC$ at vertex $A$ intersects the line $BC$ at point $A_1$ , and point $A_2$ is the midpoint of segment $A_1 A$ . Points $B_2$ and $C_2$ are defined analogously. Prove that the points $A_2, B_2,$ and $C_2$ are lying on the same line orthogonal to the Euler line of $\triangle ABC$ .
Exercise 8. Let $BB'$ and $CC'$ be the altitudes of a triangle $ABC$ and $H$ its orthocenter. Let $M$ be the midpoint of $BC$ and $D$ the intersection point of the lines $BC$ and $B'C'$ . Show that $DH$ is perpendicular to $AM$ .
Exercise 9. Assume that the incircle of a scalene $\triangle ABC$ touches the sides $BC, CA,$ and $AB$ at the points $D, E,$ and $F$ , respectively. Let $M$ be a point such that the incircle of $\triangle BCM$ touches $BC$ at $D$ , and the sides $BM$ and $CM$ at the points $P$ and $Q$ , respectively. Show that the lines $BC, EF,$ and $PQ$ are concurrent.

3. Brianchon's Theorem

The next theorem is an important result and it holds in a projective setting.

Theorem 4: Brianchon's Theorem. Let $ABCDEF$ be a tangent hexagon. Prove that the lines $AD, BE,$ and $CF$ intersect at the same point.

Exercise 10. Let $ABCD$ be a circumscribed quadrilateral whose incircle touches $AB, BC, CD,$ and $DA$ at $M, N, P,$ and $Q$ , respectively. Prove that the lines $AC, BD, MP,$ and $NQ$ are concurrent.

4. Brocard's Theorem

The following fact is yet another important projective result that can be extremely useful in olympiad problem solving.

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Proposition 5. Let $ABCD$ be a cyclic quadrilateral. Let the lines $AB$ and $CD$ intersect at $P$ , the lines $BC$ and $DA$ at $Q$ , and the diagonals $AC$ and $BD$ intersect at $R$ . Let $PX$ and $PY$ be tangents to the circumscribed circle of $ABCD$ . Then the points $Q, X, R,$ and $Y$ are collinear.

Methodological Suggestions: On some of the later training we are going to learn about the pole and the polar line that have deep meaning in projective geometry. It is essential to explain the power of a point well and emphasize its role in proving the fundamental results in geometry.

Theorem 6: Brocard's Theorem. Let $ABCD$ be a cyclic quadrilateral and $O$ its circumcenter. Let the lines $AB$ and $CD$ intersect at $P$ , the lines $BC$ and $DA$ at $Q$ , and the diagonals $AC$ and $BD$ intersect at $R$ . Then $O$ is the orthocenter of $\triangle PQR$ .

P1 IMO

Sharygin: Let $I$ be the incenter of triangle $ABC$ , and $M,N$ be the midpoints of arcs $ABC,BAC$ of its circumcircle. Prove that points $M,I,N$ are collinear if and only if $AC + CB = 3AB.$
Let $ABC$ be an acute triangle. A line perpendicular to $BC$ cuts $BC,AC,AB$ in $D,E,F$ . Prove that the orthocenters of $\triangle ABC$ and $\triangle AEF$ are collinear with $D$ .

P1+ IMO

Let $ABCDEF$ be a convex hexagon such that the triangles $ACE,BDF$ have the same centroid ( $G$ ). The points $X,Y,Z$ lie on $AD,BE,CF$ such that $\frac{AX}{DX} = \frac{EY}{YB} = \frac{CZ}{ZF}.$ Prove that the point $G$ is the centroid of $\triangle XYZ$ .
RMM Let $ABC$ be an acute triangle ( $AB = AC$ ) with incenter and circumcenter $I, O$ . Let the incircle touch $BC,AC,AB$ at $D,E,F$ . Assume that the line through $I$ parallel to $EF$ , the line through $D$ parallel to $AO$ , and the altitude from $A$ are concurrent. Prove that the concurrency point is the orthocenter of $\triangle ABC$ .

P2 IMO

Ukraine TST: Let $ABC$ be an acute triangle with circumcenter and centroid $O,G$ . The perpendicular bisectors of the segments $GA,GB,GC$ intersect at points $D,E,F$ . Prove that $O$ is the centroid of $\triangle DEF$ .
NSMO: Let $ABC$ be a triangle. The circle $(O)$ passes through the points $B,C$ and cuts $AB,AC$ again at $D,E$ . $H$ is the intersection of the segments $BE$ and $CD$ . Points $F, G$ lie on $AB,AC$ such that $AD = BF, AE = CG.$ $K$ is the circumcenter of $\triangle AFG$ . Prove that $AK \parallel HO$ .
USA TSTST: Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$ . A line through $H$ intersects segments $AB$ and $AC$ at $E$ and $F$ , respectively. Let $K$ be the circumcenter of $\triangle AEF$ , and suppose line $AK$ intersects $\Gamma$ again at a point $D$ . Prove that line $HK$ and the line through $D$ perpendicular to $BC$ meet on $\Gamma$ .

P2+ IMO

Iran TST: Let $ABC$ be a triangle with circumcenter $O$ . Points $X,Y$ lie on $AB,AC$ such that reflection of $BC$ on $XY$ is tangent to the circumcircle of $\triangle AXY$ . Prove that the circumcircle of $\triangle AXY$ is tangent to the circumcircle of $\triangle BOC$ .
Iran TST: Let $ABC$ be a triangle, arbitrary points $P, Q$ lie on side $BC$ such that $BP = CQ$ and $P$ lies between $B,Q$ . The circumcircle of $\triangle APQ$ intersects sides $AB,AC$ at $E,F$ . The point $T$ is the intersection of $EP,FQ$ . Two lines passing through the midpoint of $BC$ and parallel to $AB,AC$ intersect $EP,FQ$ at points $X,Y$ . Prove that the circumcircle of $\triangle TXY$ and $\triangle APQ$ are tangent to each other.
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$ . Circles $\omega_B$ passing through $B$ and $\omega_C$ passing through $C$ are tangent at $I$ . Let $\omega_B$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M$ , and let $\omega_C$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N$ . Rays $PM$ and $QN$ meet at $X$ . Let $Y$ be a point such that $YB$ is tangent to $\omega_B$ and $YC$ is tangent to $\omega_C$ . Show that $A, X, Y$ are collinear.

P3 IMO

Let $ABC$ be an acute triangle with centroid $G$ , orthocenter $H$ . The line passing through $G$ and the projection of $A$ on $BC$ cuts the circumcircle of $\triangle ABC$ at $A'$ . Define $B', C'$ similarly. Prove that $AA', BB', CC', GH$ are concurrent.
Ukraine TST: Let $ABC$ be an acute triangle with incircle $\Gamma$ touch sides $BC,CA,AB$ at $D,E,F$ . Let $I_a$ be the excenter opposite to $A$ in $\triangle ABC$ . Define $G$ as the centroid of $\triangle DEF$ . Let $H$ be the projection of $D$ on $EF$ . Prove that lines $GH$ and $I_a D$ intersect at a point on $\Gamma$ .

Level 4

P1 IMO

Sharygin Let $I$ be the incenter of triangle $ABC$ , and $M,N$ be the midpoints of arcs $ABC,BAC$ of its circumcircle. Prove that points $M,I,N$ are collinear if and only if $AC + CB = 3AB.$
Let $ABC$ be an acute triangle. A line perpendicular to $BC$ cuts $BC,AC,AB$ in $D,E,F$ . Prove that the orthocenters of $\triangle ABC$ and $\triangle AEF$ are collinear with $D$ .

P1+ IMO

Let $ABCDEF$ be a convex hexagon such that the triangles $ACE,BDF$ have the same centroid ( $G$ ). The points $X,Y,Z$ lie on $AD,BE,CF$ such that $\frac{AX}{DX} = \frac{EY}{YB} = \frac{CZ}{ZF}.$ Prove that the point $G$ is the centroid of $\triangle XYZ$ .
RMM Let $ABC$ be an acute triangle ( $AB = AC$ ) with incenter and circumcenter $I, O$ . Let the incircle touch $BC,AC,AB$ at $D,E,F$ . Assume that the line through $I$ parallel to $EF$ , the line through $D$ parallel to $AO$ , and the altitude from $A$ are concurrent. Prove that the concurrency point is the orthocenter of $\triangle ABC$ .

P2 IMO

Ukraine TST Let $ABC$ be an acute triangle with circumcenter and centroid $O,G$ . The perpendicular bisectors of the segments $GA,GB,GC$ intersect at points $D,E,F$ . Prove that $O$ is the centroid of $\triangle DEF$ .
NSMO Let $ABC$ be a triangle. The circle $(O)$ passes through the points $B,C$ and cuts $AB,AC$ again at $D,E$ . $H$ is the intersection of the segments $BE$ and $CD$ . Points $F, G$ lie on $AB,AC$ such that $AD = BF, AE = CG.$ $K$ is the circumcenter of $\triangle AFG$ . Prove that $AK \parallel HO$ .
USA TSTST Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$ . A line through $H$ intersects segments $AB$ and $AC$ at $E$ and $F$ , respectively. Let $K$ be the circumcenter of $\triangle AEF$ , and suppose line $AK$ intersects $\Gamma$ again at a point $D$ . Prove that line $HK$ and the line through $D$ perpendicular to $BC$ meet on $\Gamma$ .

P2+ IMO

Iran TST Let $ABC$ be a triangle with circumcenter $O$ . Points $X,Y$ lie on $AB,AC$ such that reflection of $BC$ on $XY$ is tangent to the circumcircle of $\triangle AXY$ . Prove that the circumcircle of $\triangle AXY$ is tangent to the circumcircle of $\triangle BOC$ .
Iran TST Let $ABC$ be a triangle, arbitrary points $P, Q$ lie on side $BC$ such that $BP = CQ$ and $P$ lies between $B,Q$ . The circumcircle of $\triangle APQ$ intersects sides $AB,AC$ at $E,F$ . The point $T$ is the intersection of $EP,FQ$ . Two lines passing through the midpoint of $BC$ and parallel to $AB,AC$ intersect $EP,FQ$ at points $X,Y$ . Prove that the circumcircle of $\triangle TXY$ and $\triangle APQ$ are tangent to each other.
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$ . Circles $\omega_B$ passing through $B$ and $\omega_C$ passing through $C$ are tangent at $I$ . Let $\omega_B$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M = B$ , and let $\omega_C$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N = C$ . Rays $PM$ and $QN$ meet at $X$ . Let $Y$ be a point such that $YB$ is tangent to $\omega_B$ and $YC$ is tangent to $\omega_C$ . Show that $A, X, Y$ are collinear.

P3 IMO

Let $ABC$ be an acute triangle with centroid $G$ , orthocenter $H$ . The line passing through $G$ and the projection of $A$ on $BC$ cuts the circumcircle of $\triangle ABC$ at $A'$ . Define $B', C'$ similarly. Prove that $AA', BB', CC', GH$ are concurrent.
Ukraine TST Let $ABC$ be an acute triangle with incircle $\Gamma$ touch sides $BC,CA,AB$ at $D,E,F$ . Let $I_a$ be the excenter opposite to $A$ in $\triangle ABC$ . Define $G$ as the centroid of $\triangle DEF$ . Let $H$ be the projection of $D$ on $EF$ . Prove that lines $GH$ and $I_a D$ intersect at a point on $\Gamma$ .

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