Geometric Proofs

In this note, I have a selection of questions for you to prove. It is for all levels to try. The theme is geometry. Good luck.

1) Scalene triangle $ABC$ is right-angled at $A$. The tangent to the circumcircle of triangle $ABC$ at point $A$ intersects $BC$ at $X$. Let the points of contact of the incircle of triangle $ABC$ with sides $AB$ and $AC$ be $E$ and$F$ respectively. Let $EF$ and $BC$ at $Y$ and $AX$ at $Z$.

Prove that triangle $XYZ$ is both obtuse and isosceles.

2) Let $ABCD$ be a square with centre $F$. Let $DFCE$ be a square and let $BEHG$ be the square containing $C$ in its interior.

Prove that $C$ is the midpoint of $AH$.

3) In triangle $ABC$ it is known that $\angle BAC = 2 \angle ACB$ and $2 \angle ABC = \angle BAC + \angle ACB$. The bisector of angle $ACB$ intersects $AB$ at $E$. Let $F$ be the midpoint of $AE$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. The perpendicular bisector of $DF$ intersects $AC$ at $G$.

Prove that $AG = CG$.

4) Point $D$ lies outside circle $\Gamma$. A line $\mathit{l}$ through $D$ intersects $\Gamma$ at points $E$ and $F$. Points $A$ and $B$ are the points of contact of the two tangents from $D$ to $\Gamma$. The line passing through $B$ and parallel to $\mathit{l}$ intersects $\Gamma$ at $G$.

Prove that $GA$ intersects the segment $EF$ at its midpoint.

5) Acute triangle $ABC$ has circumcircle $\Omega$. The tangent at $A$ to $\Omega$ intersects $BC$ at $D$. Let $E$ be the midpoint of the segment $AD$. Let $F$ be the second intersection point of $BE$ with $\Omega$. Let $G$ be the second intersection point of $DF$ with $\Omega$.

Prove that $CG$ is parallel to $DA$.

6) Prove that if 3 congruent circles pass through the same point, then their other three intersection points lie on a fourth circle with the same radius.