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# 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.

Note by Sharky Kesa
2 years, 8 months ago

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Hint:

1.) Angle chasing

2.) Pythagoras

6.) Consider the centroid of triangle formed by each center.

Too bad I'm not in mood of geometry right now. =..=" · 2 years, 8 months ago

Are you now? · 2 years, 8 months ago