NOTATION:
PQ denotes the vector from point P to point Q.
• denotes the vector dot product.
PROOF:
WOLOG let the circle have unit radius and O be its center.
Let
P = (AB&CD) ∩ (AC&BD)
Let M and N be the midpoints of line segments AB and AC
respectively. Therefore,
OA + OB OC + OD
OP = OM + MP = --------- + λ ---------
2 2
and
OA + OC OB + OD
OP = ON + NP = --------- + μ ---------
2 2
Thus,
(1-μ)OB + (λ-1)OC + (λ-μ)OD = 0
1-μ λ-1
λ≠μ ⇒ OD = ----- OB + ----- OC
μ-λ μ-λ
1-μ
τ = ----- ⇒ OD = τ OB - (τ+1)OC
μ-λ
1 = OD•OD = τ2 - 2τ(τ+1)OB•OC + (τ+1)2
0 = τ(τ+1)(1 - OB•OC)
τ(τ+1) = 0 ⇒ λ = 1 or μ = 1
⇒ OP = (OA+OB+OC+OD)/2
(1 - OB•OC) = 0 ⇒ B = C which contradicts
B and C as distinct
λ=μ ⇒ (1-λ)(OB - OC) = 0
(1-λ) = 0 ⇒ OP = (OA+OB+OC+OD)/2
(OB - OC) = 0 ⇒ B = C which contradicts
B and C as distinct
Therefore, OP = (OA+OB+OC+OD)/2
Let
P = (AB&CD) ∩ (AD&BC)
Then OP = (OA+OB+OC+OD)/2 by the same argument as above
swapping letters C and D.
Therefore, (AB&CD), (AC&BD), and (AD&BC) are concurrent.
QED
HOW CONCYLIC FEET PARTS 1&2 ARE RELATED:
If A, B, C, and D are distinct points on a
circle, then the point of concurrency of Part 2
is the center of the circle on which the feet of
Part 1 lie.
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