Diagrams | ELEMENTARY GEOMETRY

Triangle ABC with P on its circumcircle. The perpendicular feet D, E, F on BC, CA, and the extension of AB lie on one line.

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The Simson Line

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Acute triangle ABC with M the midpoint of BC. Two circles through M are tangent to AB at B and AC at C and meet again at D. The circumcircle of ABCD meets line DM again at H, the reflection of A across the perpendicular bisector of BC.

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Power and Midpoint Symmetry in BMO1 2025

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Triangle ABC with its circumcircle, tangent intersections P, Q, R, angle-bisector traces D, E, F, and the lines PD, QE, RF meeting at T.

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Tangents, Angle Bisectors, and a Common Point

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Triangle ABC with three similarly oriented similar isosceles triangles constructed externally, and the lines AX, BY, CZ meeting at K.

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Ceva, Kiepert, and Jacobi

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Triangle ABC with external equilateral triangles and their centers P, Q, R joined to form the equilateral Napoleon triangle.

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Napoleon and Van Aubel

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Triangle ABC with its three altitudes, circumcircle, and nine-point circle; the side midpoints, altitude feet, and vertex–orthocenter midpoints lie on the smaller circle, whose center N is the midpoint of OH.

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The Nine-Point Circle: One Homothety

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Four Orthocenters: The Half-Turn Configuration

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Three nonconcentric circles with their three pairwise radical axes meeting at one point.

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Radical Axes and Orthogonal Parabolas

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Triangle ABC with medial triangle DEF; the three altitudes of DEF meet at O, the circumcenter of ABC.

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The Euler Line: Medial Triangle and Homothety

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A scalene triangle ABC with L, M, and N at the side midpoints. The medians AL, BM, and CN meet at G.

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The Centroid: Medians and Isotomic Conjugation

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Triangle ABC with three internal angle-bisector cevians meeting at the incenter I.

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The Incenter: Trigonometric Ceva

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An acute triangle ABC with D, E, and F the perpendicular feet from A, B, and C, respectively. The altitudes AD, BE, and CF meet at H.

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Two Views of the Orthocenter

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Three Circles and an Incircle Inversion for Problem 4941

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Two directly similar triangles OAB and OCD. Lines AE and DF meet at P, while AC and BD meet at Q. The points O, Q, and P are collinear. The two circumcircles have centers K and L, and OP is perpendicular to KL.

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Pappus and Radical Axis for Problem 5021

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Triangle ABC with incircle contact points D, E, and F. A prime and C prime are the midpoints of EF and DE; P lies on DE and on the line through A parallel to EF; X is the intersection of A prime C and IP.

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Incircle Inversion for Problem 5031

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Parallelogram ABCD with BC equal to BD and M the midpoint of CD. E, F, and G are the perpendicular feet from M to the lines BD, AD, and BC. The circle centered at M passes through E, F, G, and K; AK is tangent at K, and N is the intersection of EG and BM.

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Tangent Circle in a Parallelogram for Problem 5088

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Cyclic quadrilateral BCAD with CB equal to CD. A₁ and B₁ are the altitude feet in triangle ABC; H bisects AA₁, and DB₁ is perpendicular to DB at D.

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Cyclic Orthocenter Configuration for Problem 5098

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Rectangle ABCD with M between B and C, N beyond C, perpendicular feet P and Q from B to AM and AN, and line PQ meeting BC at S and line AD at R.

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Auxiliary Rectangle and Fixed-Point Construction for Problem 5109

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