Two Circles and One More
The applet below illustrates the following problem:
Two circles C(E) with center E and C(F) with center F intersect in points P and Q. EP extended intersects again C(E) in G and C(F) in M. FP extended intersects again C(E) in L and C(F) in H. Then points E, F, L, M, Q are concyclic and |
|What if applet does not run?|
Triangles LEP and MFP are isosceles,right,obtuse,scalene,isosceles with equal base angles EPL and FPM. Therefore, their apex angles are also equal:
|∠LEP = ∠MFP.|
But ∠LEP is the same as ∠LEM and, similarly, ∠MFP is the same as ∠LFM. This makes quadrilateral LEFM cyclic,cyclic,parallelogram,rhombus,trapezoid. We'll show that Q also belongs to the circumcircle LEFM.
∠LQP is inscribed into C(E) and is subtended by the arc of the central,right,acute,central,full angle LEP. Therefore,
|∠LQP = ½∠LEP.|
|∠MQP = ½∠MFP.|
Since ∠LEP = ∠MFP, ∠LQM = ∠LQP + ∠PQM = ∠LEM = ∠LFM. It follows that quadrilaterals LEQM and LQFM are cyclic and their circumcircles coincide.
Copyright © 1996-2018 Alexander Bogomolny