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# Triangle in a Triangle

## You may also like

### Fitting In

### Triangle Midpoints

Links to the University of Cambridge website
Links to the NRICH website Home page

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30 April (Primary), 1 May (Secondary)

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Age 14 to 16

Challenge Level

*Triangle in a Triangle printable sheet*

Draw a right-angled triangle $ABC$, and mark a point $\frac13$ of the way along $AB$, $\frac13$ of the way along $BC$ and $\frac13$ of the way along $CA$.

Join your three points together to form a new triangle.

**Can you work out the fraction of the original triangle that is covered by your new triangle?**

*You may wish to explore using the interactive diagram below.*

Try a few examples. What do you notice?

Can you explain why?

*Perhaps it might help to add some extra lines... click below to see.*

We started with a right-angled triangle... what about other triangles?

What fraction of the triangle is shaded purple?

Can you prove it?

*You may wish to try Areas and Ratios and Another Triangle in a Triangle next.*

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?