Three Tangents to a Parabola
Two tangents to any parabola are cut by a third in ratios summing to one, via similarity to y = x^2.
1 Statement
Let \(T_1\) and \(T_2\) be tangents to a parabola at distinct points \(P_1\) and \(P_2\), meeting at \(P\). A third tangent \(T\), at a point other than \(P_1\) and \(P_2\), meets \(T_1\) at \(Q_1\) and \(T_2\) at \(Q_2\). Since \(Q_1\) lies on the line \(PP_1\), there is a unique real number \(t_1\) with \(Q_1 = P + t_1 (P_1 - P)\); define \(t_2\) in the same way from \(Q_2 = P + t_2 (P_2 - P)\). Thus \(\abs {t_1} = PQ_1 / PP_1\) and \(\abs {t_2} = PQ_2 / PP_2\).
Theorem 1.1 (Three tangents). In this situation \(t_1 + t_2 = 1\). If the point of contact of \(T\) lies on the arc between \(P_1\) and \(P_2\), then \(0 < t_1, t_2 < 1\), and hence
2 Setting
Definition 2.1 (Similarity). A similarity of the plane is a map \(S(X) = \lambda R X + v\), where \(\lambda > 0\), \(R\) is a rotation or reflection, and \(v \in \R ^2\).
A similarity is an affine bijection. It maps lines to lines, preserves tangency between lines and smooth curves, and multiplies every length by \(\lambda \). In particular, it preserves the coefficients \(t_1\) and \(t_2\), because \(S(P + t (P_1 - P)) = S(P) + t \bigl ( S(P_1) - S(P) \bigr )\).
Lemma 2.2 (All parabolas are similar). Every parabola is the image of \(y = x^2\) under a similarity.
Proof. A parabola with focal distance \(p > 0\) can be moved by a rotation and a translation to \(y = x^2 / (4p)\). The scaling \((x, y) \mapsto (4p\,x, 4p\,y)\) takes \(y = x^2\) to \(y/(4p) = (x/(4p))^2\), that is, to \(y = x^2/(4p)\).
By Lemma 2.2 and the invariance of \(t_1\) and \(t_2\), it suffices to prove Theorem 1.1 for \(y = x^2\).
3 Idea of the proof
On \(y = x^2\) the tangents at \(s\) and \(u\) meet at \(\bigl (\frac {s+u}{2}, su\bigr )\). Measured along a tangent line, the parameters \(t_1\) and \(t_2\) are ratios of \(x\)-coordinates, and they add to one by a single cancellation.
4 Proof
Lemma 4.1 (Tangent intersections). On \(y = x^2\), the tangent at \((s, s^2)\) is \(y = 2sx - s^2\). For \(s \neq u\), the tangents at \(s\) and \(u\) meet at \(\bigl (\frac {s+u}{2}, su\bigr )\).
Proof. The slope at \(x = s\) is \(2s\), so the tangent is \(y - s^2 = 2s(x - s)\). Two such lines meet where \(2(s - u)x = s^2 - u^2\), that is, at \(x = \frac {s+u}{2}\), and then \(y = 2s \cdot \frac {s+u}{2} - s^2 = su\).
Proof of Theorem 1.1. Let the three points of contact have \(x\)-coordinates \(a\), \(b\), \(c\), all distinct, with \(P_1 = (a, a^2)\) and \(P_2 = (b, b^2)\). By Lemma 4.1,
The line \(T_1\) is not vertical, so \(t_1\) is determined by \(x\)-coordinates alone:
In the same way, on \(T_2\),
Hence
If \(c\) lies strictly between \(a\) and \(b\), then \(c - b\) has the sign of \(a - b\) and \(\abs {c - b} < \abs {a - b}\), so \(0 < t_1 < 1\); likewise \(0 < t_2 < 1\). Then \(PQ_1/PP_1 + PQ_2/PP_2 = t_1 + t_2 = 1\).
5 Remarks
Remark 5.1. When the third point of contact lies outside the arc \(P_1P_2\), exactly one of \(t_1\), \(t_2\) is negative, and the identity becomes \(\absStar{\frac {PQ_1}{PP_1} - \frac {PQ_2}{PP_2}} = 1\). The signed form \(t_1 + t_2 = 1\) covers both cases.
Remark 5.2. Since \(t_1 = \frac {c - b}{a - b}\) depends on \(c\) linearly, the map \(Q_1 \mapsto Q_2\) from \(T_1\) to \(T_2\) is affine. A moving tangent therefore cuts two fixed tangents in proportional segments. This is the classical construction of a parabola as the envelope of lines joining corresponding points of two proportionally divided segments, as in string art.