THEOREM D: If J is any simple closed curve and T is any triangle then J has an inscribed triangle similar to T.

Proof: The proof takes place in five steps.

Step 1. Place a circle inside the curve J and move it in some direction until it contacts J.

Step 2. Let X be a point where this circle first touches J (there may be more than one such point). Inscribe a triangle XYZ in the circle so that XYZ is similar to T and YZ is a longest side. (Note that any triangle can be inscribed in a circle.) Now move Y and Z away from X proportionally (so that XYZ is similar to T at all times). Stop the motion when one of them first touches the curve J.

Step 3. If Y and Z touch J simultaneously then we are done (since XYZ would then be inscribed in J). So, assume Y touches first. Then we now have X and Y on the curve J with Z still inside of J. Choose two points P and Q on J that are at maximum distance from each other. Gradually move X along J toward point P, all the time moving Z to preserve the fact that XYZ is similar to T.

Step 4. Once X is moved to P, begin moving Y along the curve toward Q, again moving Z to preserve the similarity class of XYZ.

Step 5. Now YZ is at least as long as XY (because we YZ is a longest side of the triangle) and XY is the maximum distance possible for any two points on J. So, Z must be either outside or on the curve J. If it is on J then we are done (for XYZ would then be inscribed in J), and if it is outside J then we know that at some time during steps 3 and 4 Z must have crossed J, for it began inside the curve. At the instant Z crossed J we had our desired inscribed triangle.

Note: This is a truly pretty proof, but it merely shows there must be such a triangle somewhere. It really doesn't tell you where that triangle will occur, and using this method to actually find the triangle would be very difficult because of the simultaneous motion of points it requires. But Theorem E will show us that no matter where we are on the curve J, we wouldn't have to move far to find a vertex of one of our inscribed triangles.