Problem Set #2 - Homepage Usask

MATH 2306 History of Mathematics, Winter 2011 February 16, 2011
20. Given a triangle, inscribe a square in it so that one of the sides of the square is on the base, and the
other two vertices are on the other two sides.
Other Geometric Problems:
21. Let p and q be two different prime numbers. Suppose you know how to construct regular p-gon and
regular q-gon. Explain how to construct regular pq-gon.
22. Prove that if a quadrilateral has a circle inscribed in it, then the sum of the lengths of one pair of
opposite sides equals the sum of the lengths of the other pair of opposite sides.
23. Suppose you have a 19 ◦ -wedge. That is, you have a geometric tool that can construct 19 ◦ angles
only. In addition to that you have a pencil, i.e. you can mark any point along the sides of the wedge,
including the vertex of the angle. No other instruments are available. Construct 1 ◦ angle.
24. Using the wedge tool introduced above, can you construct 1 ◦ if the angle of the wedge is 17 ◦ ? What
if it is 18 ◦ ?
25. In ∆ABC a circle is outscribed, so that it touches the side AB and the extensions of the sides AC and
BC. Prove that the tangents to this circle from point C equal half the perimeter of ∆ABC.
26. In a convex quadrilateral, is the sum of the two diagonals smaller than, equal to, or greater than the
perimeter of the quadrilateral?
27. Same question as above, with respect to the half-perimeter.
28. Prove that in a non-isosceles triangle, the angle bisector of the angle opposite the base lies between
the median and the altitude to the base.