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A triangle T has an area of 1cm^2. Let M be the product of the perimeter of T and the sum of the three altitudes of T. Which of the following statements is false?
A. There are (or there exist) triangles T for which M> 1000
B. M> 6 for all triangles T
C. There are triangles T for which M=18
D. M> 16 for all right-angled triangles
E. There are triangles T for which M< 12
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
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?
Take any point P inside an equilateral triangle. Draw PA, PB and PC from P perpendicular to the sides of the triangle where A, B and C are points on the sides. Prove that PA + PB + PC is a constant.