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 ABCABC is right-angled at AA. The tangent to the circumcircle of triangle ABCABC at point AA intersects BCBC at XX. Let the points of contact of the incircle of triangle ABCABC with sides ABAB and ACAC be EE andFF respectively. Let EFEF and BCBC at YY and AXAX at ZZ.

Prove that triangle XYZXYZ is both obtuse and isosceles.


2) Let ABCDABCD be a square with centre FF. Let DFCEDFCE be a square and let BEHGBEHG be the square containing CC in its interior.

Prove that CC is the midpoint of AHAH.


3) In triangle ABCABC it is known that BAC=2ACB\angle BAC = 2 \angle ACB and 2ABC=BAC+ACB2 \angle ABC = \angle BAC + \angle ACB. The bisector of angle ACBACB intersects ABAB at EE. Let FF be the midpoint of AEAE. Let DD be the foot of the perpendicular from AA to BCBC. The perpendicular bisector of DFDF intersects ACAC at GG.

Prove that AG=CGAG = CG.


4) Point DD lies outside circle Γ\Gamma. A line l\mathit{l} through DD intersects Γ\Gamma at points EE and FF. Points AA and BB are the points of contact of the two tangents from DD to Γ\Gamma. The line passing through BB and parallel to l\mathit{l} intersects Γ\Gamma at GG.

Prove that GAGA intersects the segment EFEF at its midpoint.


5) Acute triangle ABCABC has circumcircle Ω\Omega. The tangent at AA to Ω\Omega intersects BCBC at DD. Let EE be the midpoint of the segment ADAD. Let FF be the second intersection point of BEBE with Ω\Omega. Let GG be the second intersection point of DFDF with Ω\Omega.

Prove that CGCG is parallel to DADA.


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
4 years, 12 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. =..="

Samuraiwarm Tsunayoshi - 4 years, 12 months ago

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Are you now?

Sharky Kesa - 4 years, 11 months ago

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Probably in a week or 2. btw Proofathon Geometry will be here in 2 days though. =="

Samuraiwarm Tsunayoshi - 4 years, 11 months ago

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