# 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
2 years, 7 months 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.

- 2 years, 7 months ago