Lemmas in Geometry
Lemmas in Geometry
Lemmas in Geometry
-Navneel Singhal
Introduction
Here are some lemmas which can be useful in Olympiad Geometry.
Most are well known and some are due to the author himself, so have
fun proving them and using them to the fullest advantage in your
Olympiad journey.
A word of warning
You can use these lemmas on the actual Olympiad only after you prove
them on the test, because you can’t quote lemmas on an Olympiad.
The Lemmas
1. If H is the orthocentre of a triangle ABC and M is the midpoint of
BC then the circle with AH as diameter, circumcircle of BHC and
AM are concurrent.
3. If the circle through vertex A and the midpoint A’ of the arc BAC
of the circumcircle of ABC cuts AB and AC at B’ and C’
respectively then BB’ = CC’.
12. The cevian triangles of isotomic conjugates have the same area.
13. If a line makes equal angles with the opposite sides of a cyclic
quadrilateral, then circles can be drawn tangent to each pair, where
this line meets them, and these circles are coaxial with the original
circle.
14. The medial triangle and the triangle homothetic to the original
triangle at the Nagel point share a common incircle.
15. Given an angle and a circle through the vertex of the angle, cutting
its bisector at a fixed point. Then the sum of the intercepts of the
circle on the sides of the angle is invariant.
19. Let C3 be a circle coaxial with 2 circles C1 and C2. Then it is the
locus of points such that the ratio of the powers of the point with
respect to C1 and C2 is constant.
20. Let Ia, Ib, Ic be the excenters and M1, M2, M3 be the midpoints of
the arcs BAC, ABC and ACB of the circumcircle. Then I, the
incentre, is the orthocentre of the excentral triangle, M1 M2M3 is
the medial triangle of the excentral triangle and IbIcBC etc are
cyclic with diameters as IbIc etc and IBIaC etc are cyclic with
diameters IIa etc respectively. The circumcircle of ABC is the nine
point circle and ABC is the orthic triangle of the excentral triangle.
Also, the contact triangle is homothetic with the excentral triangle.