# My best 8th Grade geometry problems!

Feel free to post more!

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 $$\displaystyle AM=AN$$.

2 years, 5 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$

Sort by:

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, 5 months ago