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qwerty123456asdfgzxcvb

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#1 h Apr 21, 2025, 7:23 PM • 8 Y

Y by CyclicISLscelesTrapezoid, OronSH, pineapply, kiyoras_2001, golue3120, KevinYang2.71, EpicBird08, Ciobi_

Let $\mathcal{S}$ be a logarithmic spiral centered at the origin (ie curve satisfying for any point $X$ on it, line $OX$ makes a fixed angle with the tangent to $\mathcal{S}$ at $X$ ). Let $\mathcal{H}$ be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.

Prove that for a point $P$ on the spiral, the polar of $P$ wrt. $\mathcal{H}$ is tangent to the spiral.

This post has been edited 3 times. Last edited by qwerty123456asdfgzxcvb, Apr 21, 2025, 7:33 PM

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#2 h Apr 21, 2025, 10:18 PM • 1 Y

Y by qwerty123456asdfgzxcvb

Let $\mathfrak I$ be inverflection centered at $O$ fixing $P$ , and let $\mathfrak p$ be polarity with respect to $\mathcal H$ . Then $\mathcal S$ is preserved by $\mathfrak I$ . Let $P_0$ be the tangency point of $\mathcal H$ and $\mathcal S$ . Because $\mathfrak I$ is an isoconjugation in $\triangle O\infty_{+i}\infty_{-i}$ fixing $P$ and $\mathcal H$ is a diagonal conic through $P$ in $\triangle O\infty_{+i}\infty_{-i}$ , the points $P$ and $\mathfrak I(P)$ are always conjugate in $\mathcal H$ . Now we claim the polar of a point $P\in\mathcal S$ is the tangent at $\mathcal S$ to $\mathfrak I(P)$ . If $T_{\mathfrak I(P)}$ is the tangent, then $\arg O\mathfrak I(P)-\arg T_{\mathfrak I(P)}$ is constant, $\arg O\mathfrak I(P)+\arg OP$ is constant, and $\arg OP+\arg \mathfrak p(P)$ is constant, so the angle between $T_{\mathfrak I(P)}$ and $\mathfrak p(P)$ is constant. This angle is $0$ when $P=P_0$ , and hence $T_{\mathfrak I(P)}$ and $\mathfrak p(P)$ are parallel. They both pass through $\mathfrak I(P)$ , and hence coincide.

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