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Funny easy transcendental geo
qwerty123456asdfgzxcvb   0
2 hours ago
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.
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qwerty123456asdfgzxcvb
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steven_zhang123   3
N Mar 30, 2025 by steven_zhang123
In the plane, there are two line segments $AB$ and $CD$, with $AB \neq CD$. Prove that there exists and only exists one point $P$ such that $\triangle PAB \sim \triangle PCD$.($P$ corresponds to $P$, $A$ corresponds to $C$)
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steven_zhang123
Mar 30, 2025
steven_zhang123
Mar 30, 2025
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steven_zhang123
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steven_zhang123
#1 h  Mar 30, 2025, 2:29 AM
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In the plane, there are two line segments $AB$ and $CD$, with $AB \neq CD$. Prove that there exists and only exists one point $P$ such that $\triangle PAB \sim \triangle PCD$.($P$ corresponds to $P$, $A$ corresponds to $C$)
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This post has been edited 2 times. Last edited by steven_zhang123, Mar 30, 2025, 4:21 AM
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lyllyl
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lyllyl
#2 h  Mar 30, 2025, 2:59 AM
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I think the problem is wrong.For example,assumn A:(-1,2) B:(-1,0) C:(2,2) D:(4,1) So P can be (1,0) and (0.2, 4.4),not only one!
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whwlqkd
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whwlqkd
#3 h  Mar 30, 2025, 9:23 AM
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Isn’t it Miquel Point?
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steven_zhang123
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steven_zhang123
#4 h  Mar 30, 2025, 12:11 PM
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@above, yes. But do you have a solution that doesn't require the use of Miquel point and circles?
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