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A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

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Fitting In

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

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No Right Angle Here

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

Altitude Inequalities

Stage: 4 Short Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Let the sides of triangle $T$ have lengths $a$, $b$ and $c$ and the corresponding altitudes have lengths $H_a$, $H_b$ and $H_c$.

By the triangle inequality, we have $a+b > c$, $b+c > a$ and $c+a > b$ and so $a+b+c > 2c$, $a+b+c > 2a$ and $a+b+c > 2b$.

Also, since $T$ has area $1$, we have $\frac{1}{2}aH_a=1$, $\frac{1}{2}bH_b=1$ and $\frac{1}{2}cH_c=1$ and so $aH_a=2$, $bH_b=2$ and $cH_c=2$. $$M=(a+b+c)(H_a+H_b+H_c)=(a+b+c)H_a+(a+b+c)H_b+(a+b+c)H_c$$ $$ > 2aH_a+2bH_b+2cH_c=4+4+4=12$$ so $M> 12$ and hence statement E is false.

This problem is taken from the UKMT Mathematical Challenges.

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