Radical axis problems

by henderson, Sep 27, 2016, 6:05 PM

$\bf\color{blue}Radical\ axis \ problems \$
1. $C_1$ and $C_2$ are two disjoint circles. $AB$ is an external common tangent and $CD$ is an internal common tangent to them, such that $C$ is nearer to $AB$ than $D$ . Suppose $P$ is the pole of $BC$ wrt $C_2$ and $M$ is the intersection point of the polars of $P$ wrt $C_1$ and $C_2.$ Prove that the midpoint of $PM$ lies on the radical axis of $C_1$ and $C_2.$

2. Let $ABC$ be a triangle, let ${A}'$ , ${B}'$ , ${C}'$ be the orthogonal projections of the vertices $A$ , $B$ , $C$ on the lines $BC$ , $CA$ and $AB$ , respectively, and let $X$ be a point on the line $A{A}'$ . Let $\gamma_{B}$ be the circle through $B$ and $X$ , centred on the line $BC$ , and let $\gamma_{C}$ be the circle through $C$ and $X$ , centred on the line $BC$ . The circle $\gamma_{B}$ meets the lines $AB$ and $B{B}'$ again at $M$ and ${M}'$ , respectively, and the circle $\gamma_{C}$ meets the lines $AC$ and $C{C}'$ again at $N$ and ${N}'$ , respectively. Show that the points $M,$ ${M}',$ $N$ and ${N}'$ are collinear.

3. The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC, CA, AB$ at $A_1 , B_1 , C_1$ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2$ . The line $B_1 C_1$ meets the line $BC$ at $A_3$ and the line $A_2 A_3$ meets the circumcircle of $ABC$ at $A_4 (\ne A_2 ) .$ Define $B_4$ and $C_4$ similarly. Prove that the lines $AA_4 , BB_4 , CC_4$ are concurrent.

4. Let $ABC$ be a triangle. Point $D$ lies on its sideline $BC$ such that $\angle CAD = \angle CBA.$ Circle $(O)$ passing through $B,D$ intersects $AB,AD$ at $E,F$ , respectively. $BF$ meets $DE$ at $G$ . Denote by $M$ the midpoint of $AG.$ Show that $CM\perp AO.$

5. Consider a semicircle of center $O$ and diameter $AB.$ A line intersects $AB$ at $M$ and the semicircle at $C$ and $D$ such that $MC>MD$ and $MB<MA.$ The circumcircles of $\triangle AOC$ and $\triangle BOD$ intersect again at $K.$ Prove that $MK\perp KO.$ $($ My solution $)$

6. Let $ABC$ be a triangle. Let $S$ be the circle through $B$ tangent to $CA$ at $A$ and let $T$ be the circle through $C$ tangent to $AB$ at $A.$ Circles $S$ and $T$ intersect at $A$ and $D.$ Let $E$ be the point where $AD$ meets the circumcircle of $\triangle ABC.$ Prove that $D$ is the midpoint of $AE.$ $($ My solution $)$

7. Let $A,B,C$ and $D$ be four points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y.$ The line $XY$ meets $BC$ at $Z.$ Let $P$ be a point on the line $XY$ other than $Z.$ The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M,$ and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N.$ Prove that the lines $AM, DN$ and $XY$ are concurrent.

8. Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$ . Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$ . Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$ .

9. Circles $W_1,W_2$ intersect at $P,K.$ $XY$ is a common tangent of two circles which is nearer to $P$ and $X$ is on $W_1$ and $Y$ is on $W_2.$ $XP$ intersects $W_2$ for the second time in $C$ and $YP$ intersects $W_1$ in $B.$ Let $A$ be intersection point of $BX$ and $CY.$ Prove that if $Q$ is the second intersection point of the circumcircles of $\triangle ABC$ and $\triangle AXY,$ then
$\angle QXA=\angle QKP.$
10. An acute-angled triangle $ABC$ is given in the plane. The circle with diameter $\, AB \,$ intersects altitude $\, CC' \,$ and its extension at points $M$ and $N,$ and the circle with diameter $\, AC \,$ intersects altitude $\, BB' \,$ and its extension at $P$ and $Q.$ Prove that the points $M, N, P, Q$ lie on a common circle. $($ My solution $)$

This post has been edited 5 times. Last edited by henderson, Oct 2, 2016, 1:05 PM