Let-K-L-M-and-N-be-the-midpoints-of-the-sides-AB-BC-CD-and-DA-respectively-of-a-cyclic-quadrilateral-ABCD-Prove-that-the-orthocenters-of-the-triangles-AKN-BKL-CLM-and-DMN-are-the-vertices-of-

Tinku Tara June 4, 2023 Geometry 0 Comments

Question Number 16074 by Tinkutara last updated on 17/Jun/17

Let K, L, M and N be the midpoints of the sides AB, BC, CD and DA, respectively, of a cyclic quadrilateral ABCD. Prove that the orthocenters of the triangles AKN, BKL, CLM and DMN are the vertices of a parallelogram.

$$\mathrm{Let}\:{K},\:{L},\:{M}\:\mathrm{and}\:{N}\:\mathrm{be}\:\mathrm{the}\:\mathrm{midpoints}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{sides}\:{AB},\:{BC},\:{CD}\:\mathrm{and}\:{DA}, \\ $$$$\mathrm{respectively},\:\mathrm{of}\:\mathrm{a}\:\mathrm{cyclic}\:\mathrm{quadrilateral} \\ $$$${ABCD}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{orthocenters} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{triangles}\:{AKN},\:{BKL},\:{CLM}\:\mathrm{and} \\ $$$${DMN}\:\mathrm{are}\:\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{parallelogram}. \\ $$

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Let-K-L-M-and-N-be-the-midpoints-of-the-sides-AB-BC-CD-and-DA-respectively-of-a-cyclic-quadrilateral-ABCD-Prove-that-the-orthocenters-of-the-triangles-AKN-BKL-CLM-and-DMN-are-the-vertices-of-

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