Definition D1
The circle with center O and radius r is the set of all points P with OP = r. A line ℓ meeting this circle at exactly one point is called a tangent, and that one point the point of tangency.
Proof.
(1)The point of tangency T is on the circle, so OT = rA1, convention
(2)If a point P of ℓ differs from T, then P is not on the circle, so OP ≠ rA1, convention
∎
Lemma L1
A tangent ℓ does not pass through the center O.
Proof.
(1)Suppose O ∈ ℓ.assumption
(2)A line through the center meets the circle at two distinct points, so ℓ meets the circle at two points.(1), A2
(3)ℓ meets the circle at exactly one point, so this is a contradiction.(2), D1
(4)Hence O ∉ ℓ(3), R
∎
Theorem T1
For a tangent ℓ with point of tangency T, OT ⊥ ℓ
Proof.
(1)O ∉ ℓ, so the foot H of the perpendicular from O to ℓ exists uniquely.L1, A3
(2)Suppose H ≠ T.assumption
(3)H splits ℓ into two rays. On the ray not containing T, take a point T′ with HT′ = HT.(1), (2), A5, A6
(4)T′ is on ℓ, HT = HT′ > 0, and T and T′ are on opposite sides of H, so T′ ≠ T(2), (3), A5, A12
(5)H is the foot of the perpendicular from O to ℓ, and T, T′ are points of ℓ, so ∠OHT = ∠OHT′ = 90°(1), (3), (4), A4
(6)OH is common, HT = HT′, and the included angles are equal, so △OHT ≅ △OHT′(3), (5), A7
(7)By congruence, OT′ = OT = r(6), A8, D1
(8)T′ is on the circle and on ℓ and differs from T, so ℓ meets the circle at two points.(4), (7), A1
(9)ℓ meets the circle at exactly one point, so this is a contradiction. Hence H = T(8), D1, R
(10)H is the foot of the perpendicular from O to ℓ, so OT ⊥ ℓ(1), (9)
∎
Theorem T2
The line ℓ through a point T of the circle perpendicular to OT is a tangent to the circle, and T is its point of tangency.
Proof.
(1)T is a point of the circle, so OT = rassumption, A1
(2)Let P be a point of ℓ other than T. OT ⊥ ℓ and T is the foot of that perpendicular, so ∠OTP = 90°assumption, A4
(3)The hypotenuse of the right triangle OTP is OP, so OP² = OT² + TP²(2), A9
(4)P ≠ T, so TP > 0 and TP² > 0, and OP² > OT² = r²(1), (2), (3), A12, A10
(5)Lengths are nonnegative, so OP > r(4), A11, A12
(6)OP ≠ r, so P is not on the circle. That is, ℓ meets the circle only at T.(5), A1
(7)Hence ℓ is a tangent and T is the point of tangency.(6), D1
∎