Wikipedia:Reference desk/Archives/Mathematics/2013 April 13

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April 13

A problem on Computational Geometry

I am quite not sure which field the problem bears to but i guess it is Computational Geometry.

1) Prove that the circles constructed on two sides of a triangle as diameters cover the entire triangle.

I tried and i think i have got the proof for right and obtuse triangle but cannot solve it for acute triangle.

2) Prove that a regular polygon cannot have more than three acute angles.Solomon7968 (talk) 16:38, 13 April 2013 (UTC)

I numbered your Q's for ease of response:

2) Do you mean a regular polygon ? Because I can easily think of polygons with more than 3 acute interior or exterior angles:

 ____________
 \          /   
  \/\/\/\/\/

StuRat (talk) 17:04, 13 April 2013 (UTC)

yes i mean regular one and i have edited the question to label regular Solomon7968 (talk) 17:08, 13 April 2013 (UTC)

I'm betting that the question actually referred to convex polygons, because the statement is true in that case, and a lot more interesting. The usual proof is based on the fact that if you take each angle of a convex polygon and subtract it from 180 degrees, the results have to add up to 360 degrees. Looie496 (talk) 20:02, 14 April 2013 (UTC)

These are simple Euclidean geometry problems — nothing to do with computers or numerical work. The latter is true for all convex polygons, if that helps. --Tardis (talk) 17:11, 13 April 2013 (UTC)

For a purely geometric (i.e. no computation required) approach to (1), think about the following. If the original triangle is ABC and the circles have diameters AB and AC, draw a line from corner A that is perpendicular to the side BC. Suppose this line meets BC at point D. Then we have divided the original triangle ABC into two right angled triangle ADB and ADC. Can you show that each of these two right angled triangles is covered by one of the two circles ? Gandalf61 (talk) 11:56, 14 April 2013 (UTC)

1) Say your triangle is $\triangle ABC$ and you've selected AB and AC as circles' diameters. Project A orthogonally on the BC line – that would be point D. Now D is a right angle both in $\triangle ABD$ and $\triangle ACD$. That means – by Thales theorem (or may be its inverse) – that each of the smaller triangles is covered by the respective circle. --CiaPan (talk) 19:13, 14 April 2013 (UTC)