[b]Proposition:[/b] [i]To draw three circles in a triangle, each of which is tangent to the other two, and to two sides of the given triangle.[/i]
Step-by step solution using the Construction Protocol.
To me, the beauty of this problem is the elegance of the geometric transformations. Algebraically, the constraints are a nest of time-consuming difficulties. For example, try solving for the radii directly from the equations of the last worksheet, without rearranging the problem by way of the sin² or cos² of auxiliary angles. It can be done, but it's your afternoon.
Geometrically, we can give the solution to ourselves. Consider the following journey, which is relatively easy:
[i]Begin at a vertex. Follow the angle bisector to O, and hop on a perpendicular to one side. The perpendicular divides the side according to the interior, half angles of the two nearest vertices.[/i]
The solution to Malfatti's problem is to turn the journey inside out. Return to the vertex, and go in the reverse order, [i]outside the trianagle:[/i] Follow a perpendicular to one of three equal circles....
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Malfatti's Problem
1. Identify and describe the constraints: [url]http://www.geogebratube.org/material/show/id/32079[/url]
[b]→2. Solution[/b]
3. Trig Supplement: [url]http://www.geogebratube.org/material/show/id/31985[/url]
This is problem #30 in Heinrich Dorrie's [i]100 Great Problems of Elementary Mathematics.[/i]
