Absolutely correct. With your invaluable hint/assistance/explanation, I now understand.
I simply needed to draw all those triangles separately and label stuff properly and I got it all sorted out and got it solved. I'll look up the numbers at some point; it was one of the last questions in this year's national Mathcounts sprint round. A student I've been working with (who got 24 right on that round, so he/she's no slouch!) showed it to me, and neither of us, nor anyone on his team, nor his coach, was able to figure it out, so thanks.
I attach a PNG picture that I drew with Geometer's Sketchpad.
A lot of things are equal in length: EO and FO are equal radii, as you said. In addition, BG and CH are congruent altitudes in the trapezoid (or are opposite sides of a rectangle). Triangle EOA is similar to triangle GBA since they both contain right angles and angle A (Angle-Angle Similarity Postulate in some books). Similarly, triangle FDO ~ HDC, with the vertices lined up by me with care.
I thin get, all with reference to the similar triangles and substitution (work it out however you wish):
AO / AB = AE / AG = EO/BG = FO/CH = DO/CD = DF / DH
Of all that, I only need the first part and the fifth part, and I get exactly your result.
Now, this suggests to me a follow-up question:
As I was making the GSP diagram, I noticed I was unable to **construct** it easily using the equivalent of Euclid's axioms that is built into GSP. I made a drawing that looks similar to the real, bonafide sitution, but if you look carefully, you may notice that point F was not really made by me as the single intersection/point of tangency of the circle. I got it as close as I could by fudging it; but the rest is all correct; that is, I made BC and AD really parallel, and BG and CH are really sides of a rectangle, since they are altitudes of the trapezoid. And the circle really is constructed to be tangent to side AD.
But point F is just a kluge, and approximation.
Here is the question: Prescribe the equivalent of a geometric construction so that point F will be in fact the single point of tangency of the cicle and leg CD. You're allowed to use almost anything in the GSP (or similar program) armory of geometric tools. I think I have it figured out. The test will be if it 'falls apart' if I move the sides around.
________________________________ From: "Lindsey, Dr. Charles" <email@example.com> To: Guy Brandenburg <firstname.lastname@example.org>; mathedcc list <email@example.com>; DC Council of Teachers of Math <firstname.lastname@example.org>; mathtalke <email@example.com> Sent: Wednesday, May 22, 2013 10:03 AM Subject: RE: a question on circles & trapezoids
Suppose you have trapezoid ABCD, with long base AD, short base BC, and non-parallel sides AB, CD. Suppose a circle has center O lying on AD, and is tangent to both AB and CD. I think I can show that AB/CD = AO/OD (since radii of the circle are perpendicular to AB and to CD, if you drop altitudes from B and C onto AD, there are at least two pairs of similar triangles). Using this, and the fact that AO+OD=AD, it should be possible to find AO and OD if the lengths of all four sides of the trapezoid are known. My argument requires only knowledge that radii of circles are perpendicular to tangent lines (at the point of tangency), and basic properties of similar triangles.
If I get time later, I will write something more detailed for review.
Chuck Lindsey, Ph.D. firstname.lastname@example.org Assistant Dean and Associate Professor of Mathematics College of Arts and Sciences Florida Gulf Coast University 10501 FGCU Blvd South, Fort Myers, FL 33965-6565 Phone: (239) 590-7168 Fax: (239) 590-7200
From:email@example.com [mailto:firstname.lastname@example.org] On Behalf Of Guy Brandenburg Sent: Tuesday, May 21, 2013 10:30 PM To: mathedcc list; DC Council of Teachers of Math; mathtalke Subject: a question on circles & trapezoids
from the latest national Mathcounts competition comes a question I have no idea how to attack:
You are given a scalene trapezoid whose side lengths I don't recall.
A circle with its center on the longer base is tangent to the two legs but not to the other base.
How on earth would you calculate the distance from the center of the circle to either end of that base?