Difference between revisions of "1992 OIM Problems/Problem 5"
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Given the height <math>h</math> in blue and length <math>m</math> in red, we begin by bisecting with straight edge and compass the line of length <math>m</math> and then bisecting it again as shown in the drawing above. We also bisect the blue line. Using the center of the blue line we measure <math>h/2</math> with the compass and draw a circle at the center of the point of the second bisection of the red line as shown. Then we can construct the rhombus by joining the intersection of the second perpendicular bisector with the circles and the edge of red line with the middle as shown above. | Given the height <math>h</math> in blue and length <math>m</math> in red, we begin by bisecting with straight edge and compass the line of length <math>m</math> and then bisecting it again as shown in the drawing above. We also bisect the blue line. Using the center of the blue line we measure <math>h/2</math> with the compass and draw a circle at the center of the point of the second bisection of the red line as shown. Then we can construct the rhombus by joining the intersection of the second perpendicular bisector with the circles and the edge of red line with the middle as shown above. | ||
| − | We then measure the length of the rhombus with the compass and use that as a radius: | + | We then measure the length of the side of the rhombus with the compass and use that as a radius: |
[[File:1992_OIM_P5c.png]] | [[File:1992_OIM_P5c.png]] | ||
Revision as of 00:11, 20 December 2023
Problem
The circumference 
and the positive numbers 
and 
are given so that there is a trapezoid 
inscribed in , of height and in which the sum of the bases 
and 
is . Build the trapezoid .
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
First we realize that the trapezoid is isosceles because it is inscribed in a circle. If we draw a rhombus using the midpoints of the trapezoid, the diagonals of the rhombus will be 
and We can construct such rhombus. Then, with similar triangles and the midpoints we know that the diagonal of the trapezoid is twice the length of the rhombus. Now we have all we need to start constructing:
Given the height in blue and length in red, we begin by bisecting with straight edge and compass the line of length and then bisecting it again as shown in the drawing above. We also bisect the blue line. Using the center of the blue line we measure 
with the compass and draw a circle at the center of the point of the second bisection of the red line as shown. Then we can construct the rhombus by joining the intersection of the second perpendicular bisector with the circles and the edge of red line with the middle as shown above.
We then measure the length of the side of the rhombus with the compass and use that as a radius:
As shown above, we draw a line and using the compass with the radius of the rhombus we draw a circle with center at any point on that line so that the circle cuts the line in two places. This will be the diameter of the circle and the length of the diagonal of the trapezoid. With the compass, we measure this diagonal with the compass and use it on the given circle 
shown in green below:
We pick point 
on the circle shown in green and using the compass with diagonal measurement we draw an arc using as center shown in magenta, so that it cuts the green circle. This intersection is . Using the given height as a radius on the compass we draw circles at and as shown. As shown above, from 
we draw a line tangent to the opposite circle using straight edge. The intersection of this line with the green circle is 
Likewise from we draw a line tangent to the opposite circle. Its intersection with the green circle is . We then draw lines connecting 
and and our trapezoid with the conditions and the given numbers is constructed.
- Note. I actually competed at this event in Venezuela when I was in High School representing Puerto Rico. I'm proud to say that I got full points on this one and I solved it very quickly. The only one I got full points. I had a straight rule and compass kit which I used to solve it as we're supposed to build the trapezoid with it and described the steps and how. Last week, after over 3 decades, I attempted this and spent a full 3 hours on it and couldn't solve it nor I remember what I did. Then I wrote this solution but I don't know if this was what I did back then.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
