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My best 8th Grade geometry problems!

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  1. A square \( \displaystyle ABCD \) has \( \displaystyle E, F, G \) are the midpoints of \( \displaystyle AD, DC, AC \). \( \displaystyle GF \) intersects \( \displaystyle AC \) at \(H\). Prove \( \displaystyle EH \perp BH \).

  2. A square \( \displaystyle ABCD \) has \( \displaystyle E, F, G, H \) lies on \( \displaystyle AB, BC, CD, AD \) so that \( \displaystyle AE=BF=CG=HD \). Prove \( \displaystyle EG \perp HF \).

  3. A rectangle \( \displaystyle ABCD \) has \( \displaystyle BH \perp AC \). \( \displaystyle K, E \) are the midpoints of \( \displaystyle AH, CD \). Prove \( \displaystyle BK \perp KE \).

  4. Acute \( \displaystyle \triangle ABC \) with orthocenter \( \displaystyle H \), \( \displaystyle M, N \) lies on \( \displaystyle BH, CH \) so that \( \displaystyle \angle AMC=\angle ANB=90° \). Prove that \( \displaystyle AM=AN \).

Note by Adam Phúc Nguyễn
10 months, 1 week ago

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2) \(\displaystyle\triangle AEH ,\displaystyle\triangle BFE ,\displaystyle\triangle CGF ,\displaystyle\triangle DHG \) are all congruent.Hence, \(EF=FG=GH=HE\) which implies \(EFGH\) is a rhombus whose diagonals intersect at right angles. Siddharth Singh · 10 months, 1 week ago

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