1999 JBMO Problems/Problem 4

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Problem 4

Let $ABC$ be a triangle with $AB=AC$. Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$).


Solution

Its easy to see that $B'$, $C'$, $D$ are collinear (since $\angle B'DC = \angle C'DC$ = 90^\circ$). Applying the sine rule in triangle$ABC$, we get$\frac{\sin BAD }{ \sin CAD} = \frac{BD }{ DC}.$Since$BAB'D$and$CC'AD$are cyclic quadrilaterals,$\angle BAD \angle BB'D$and$\angle CAD = \angle CC'D.$So,$\frac{\sin(BB'D)}{\sin(CC'D)} = \frac{BD}{DC}$and$\frac{BD}{\sin BB'D} = \frac{DC }{ \sin CC'D}.$Thus,$BB' = CC'$(the circumcircles$\mathcal{C}_1,\mathcal{C}_2$are congruent).

From right triangles$ (Error compiling LaTeX. ! Missing $ inserted.)BB'A$and$CC'A$, we have <cmath>AC'^{2} = CC'^{2} - AC^{2} = BB'^{2} - AB^{2} = AB'^{2}</cmath> So$AC' = AB'.$Since$M$is the midpoint of$B'C'$,$AM$is perpendicular to$B'C'$and hence$AM$is parallel to$BC$. So area of$[MBC] = [ABC]$and hence is independent of position of$D$on$BC$.

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