Many Circles

Geometry Level pending

Given a triange $ABC$ and three distinct points $X,Y,Z$ on $BC, AC, AB$ respectively, construct the circumcircles of $\triangle AYZ, \triangle BXZ, \triangle CXY$ and call these circles $\alpha, \beta, \gamma$ respectively. Let the intersection of circles $\alpha - \beta, \alpha - \gamma, \beta - \gamma$ be $P, Q, R$ with $P,Q,R \neq X,Y,Z$ . Find $\frac{[PQR]}{[ABC]}$ .

The answer is 0.

1 solution

Alan Yan
Nov 6, 2015

By $\textbf{Miquel's Theorem}$ , points $P,Q,R$ concur. Thus, the area is $0$ .

$\textbf{Proof}$ . Let the intersection of circles $AYZ$ and $BXZ$ be $P \neq Z$ . It is easy to see due to cyclic quads that $\angle YPZ = 180^{\circ} - \angle A$ and $\angle ZPX = 180^{\circ} - \angle B$ . This implies that $\angle YPX = 180^{\circ} - \angle C$ which gives us that quadrilateral $CYPX$ is cyclic. This tells us that all of the circles intersect at one point.

Many Circles

The answer is 0.

1 solution

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