isosceles wanted, 2 circles passing two vertices of a triangle tangent to one

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isosceles wanted, 2 circles passing two vertices of a triangle tangent to one

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Source: 2011 Ukraine MO grade XI P5

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#1 h Jul 14, 2020, 7:55 AM

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A circle that passes through the vertices $A$ and $B$ of the triangle $ABC$ , touches the side $BC$ at point $B$ and again crosses the side $AC$ at point $E$ . A second circle passes through the vertices $A$ and $C$ , touches the side $BC$ at point $C$ and crosses the side $AB$ at point $D$ . The segments $BE$ and $CD$ intersect at point $F$ . Prove that $\vartriangle BCF$ is isosceles.

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#2 h Jul 14, 2020, 4:59 PM

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parmenides51 wrote:

A circle that passes through the vertices $A$ and $B$ of the triangle $ABC$ , touches the side $BC$ at point $B$ and again crosses the side $AC$ at point $E$ . A second circle passes through the vertices $A$ and $C$ , touches the side $BC$ at point $C$ and crosses the side $AB$ at point $D$ . The segments $BE$ and $CD$ intersect at point $F$ . Prove that $\vartriangle BCF$ is isosceles.

Not that it is $\triangle BFC$ , not $\triangle BCF$ .

And here is my solution:

Let $G$ be the point diametrically opposite $C$ , then $\angle FBC=\angle EBC=\angle BAE=\angle DAC=\angle DGC= \angle DCB=\angle FCB$ So $\overline {FB}=\overline {FC}$ .

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