My best 8th Grade geometry problems!

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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 $\displaystyle EH \perp BH$ .
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 $\displaystyle EG \perp HF$ .
A rectangle $\displaystyle ABCD$ has $\displaystyle BH \perp AC$ . $\displaystyle K, E$ are the midpoints of $\displaystyle AH, CD$ . Prove $\displaystyle BK \perp KE$ .
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 $\displaystyle AM=AN$ .

#Geometry

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`2 \times 3`	$2 \times 3$
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Comments

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 - 5 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$