Geometry Stumper

[Albert Chan]

Just downloaded XCas 1.4.9-57 (win32) today.
I want to see if XCas can handle the tough proof: http://www.hpmuseum.org/forum/thread-112...#pid102245
Prove point D, C, E lies on a straight line (using distance, not slope)

pointE := head(solve([y=ae*x, y=be*(x-1)], [x, y])) //==> [-be/(ae-be), -ae*be/(ae-be)]
pointD := simplify(subst(pointE, [ae=-1/ae, be=-1/be])) //==> [-ae/(ae-be), -1/(ae-be)]
pointC := [k, 1]
dist(x) := sqrt(dot(x, x))

soln := head(solve(dist(pointD - pointC) + dist(pointE - pointC) = dist(pointD - pointE), k))

XCas was unable to solve for k, the mess is too much, *STOP* after 5 minutes ...

To help XCas a bit, scale the distance by abs(ae - be):
(ae, be was slope of line AE, BE. Since A != B, scaling factor > 0)

dist(x) := sqrt(dot(x, x) * (ae - be)^2) // scaled distance

With scaled distance, solving take 2 seconds:

soln = (-ae^2 * be - ae^2 + be^2 - be) / (ae * be^2 - ae^2 * be + ae - be)

Removing the square terms, we get:

simplify(subst(soln, [ae^2=1-2*k*ae, be^2=1-2*(k-1)*be])) //==> k

For any value of k, DCE lies on a straight line, thus DCE is always straight.