equilateral triangles on sides of equilateral, locus, equal segments

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equilateral triangles on sides of equilateral, locus, equal segments

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Source: 2021 Olympiad YuMSh Finals X p1 - Olympiad of Youth Mathematical School of St. Petersburg State University

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#1 h May 28, 2021, 10:41 PM

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An equilateral triangle $ABC$ is given. Points $X$ and $Y$ on sides $AB$ and $BC$ , respectively. Equilateral triangles $AXD$ and $BY E$ are built in the outside wrt the original triangle $ABC$ . Point $F$ is such that triangle $DEF$ is equilateral, and points $F$ and $C$ lie on the same side wrt DE.

1. It is known that $AX = BY$ , and G is the intersection point of the side $AB$ and the line passing through $F$ parallel to $BC$ . Prove that $GF = AB$ .

2. It is known that $AB$ is parallel to $FC$ . Prove that $AX = BY$ .

3. Let the initial triangle $ABC$ be fixed and its side equal to $a$ , and the rest of the points are not fixed. Let $F'$ be a point different from $F$ such that the triangle $DEF'$ is equilateral. Find locus of points F' and its length.

4. Let G be the point of intersection of the side $AB$ and the line passing through $F$ parallel to $BC$ . Prove that $BG = AX$ .

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