Questions tagged [power-of-the-point]

The power of a point $P$ with respect to a circle centered at $O$ is a measure of distance from the point to the circle, defined by $h = OP^2 - r^2$.

The power of a point $P$ with respect to a circle centered at $O$ is a measure of distance from the point to the circle, defined by

$$h = OP^2 - r^2$$

which implies that points

inside the circle have negative power
on the circle have zero power
outside the circle have positive power

For points outside the circle, the power is equivalent to the square of the length of its tangent to the circle. It is also the square of the radius of an [[wiki-orthogonal|Orthogonal]] circle centered at $P$. Unfortunately, points inside the circle have no particularly good geometric interpretation.

The power of a point is used extensively in computational geometry, including in defining the [[wiki-radical-axis-of-2-circles|radical axis]] (and thus [[wiki-radical-center-of-3-circles|radical center]]). It is better known for the power of a point theorem, which writes the power of a point in two different ways to show the equality between them. There are two cases to this theorem; the [[wiki-two-secants-2|two secants]] form and the [[wiki-tangent-secant-2|tangent-secant]] form.

53 questions

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1 answer

Let $AD\cap (BFC) $ in points $P$ and $Q$ and let $AD\cap (ABE)=M$ then $MP=MQ$.

Let $\triangle DEF$ be the medial triangle of $\triangle ABC$ with standaring notations. Let $AD\cap (BFC) $ in points $P$ and $Q$ and let $AD\cap (ABE)=M$ then $MP=MQ$. Here is the diagram: There was a non synthetic solution given here…

asked Aug 12 '20 at 06:09

Sunaina Pati

4,245

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5 answers

CGMO 2020: Prove that $X, P, Q, Y$ are concyclic.

In the quadrilateral $ABCD$, $AB=AD$, $CB=CD$, $\angle ABC =90^\circ$. $E$, $F$ are on $AB$, >$AD$ and $P$, $Q$ are on $EF$($P$ is between $E, Q$), satisfy $\frac{AE}{EP}=\frac{AF}{FQ}$. $X, Y$ are on $CP, CQ$ that satisfy $BX \perp CP, DY \perp…

geometry contest-math euclidean-geometry geometric-transformation power-of-the-point

asked Aug 10 '20 at 15:25

Sunaina Pati

4,245

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3 answers

To prove $EA = FB$ or that $CQ'$ is radical axis

Given disjoint circles $c_1 = \odot(P,PA), c_2 = \odot(O,OB)$ such that $B$ and $A$ are in the same half-plane wrt $OP$ and that $PA \parallel OB \perp OP$. Line $CDQ$ is the perpendicular bisector of $AB$, $D \in AB,Q \in OP$. Point $Q'$ is the…

geometry euclidean-geometry geometric-transformation homothety power-of-the-point

asked Nov 01 '20 at 20:25

hellofriends

2,048

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2 answers

Why is the "Power of a Point" of an Internal Point on a Circle Negative?

The Wikipedia Article on Power of a Circle gives the following explanation: In elementary plane geometry, the power of a point is a real number $h$ that reflects the relative distance of a given point from a given circle. Specifically, the power of…

circles power-of-the-point

asked Oct 26 '19 at 06:16

Vishnu

1,892

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3 answers

Prove concyclic points in a parallelogram

Let $ABCD$ be a parallelogram with obtuse angles at $A$ and $C$. Let $H$ be the foot of the perpendicular line from $A$ to $BC$, and a median from $C$ to $AB$ is drawn that extends to intersect the circumcircle of $\triangle ABC$ at $K$. Prove that…

geometry euclidean-geometry triangles circles power-of-the-point

asked Apr 30 '25 at 02:54

Poeny

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2 answers

Finding the length of $DE$ given $AB = 4$ and $BE = 5$

As shown in the diagram, $ABCD$ is a parallelogram where $DC$ is tangent to the circumcircle of $\triangle ABC$ which intersects $AD$ at $E.$ If $AB = 4$ and $BE = 5,$ find the length of $DE.$ Firstly, I noted that Power of Point could be used in…

geometry power-of-the-point

asked Dec 19 '20 at 04:24

questionasker

1,171
6
20

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2 answers

$4$ points in order $A,B,C,D$ lie on a circle with the extension of $AB$ meeting the extension of $DC$ at $E$ and that of $AD$ and $BC$ at $F$.

$4$ points in order $A,B,C,D$ lie on a circle with the extension of $AB$ meeting the extension of $DC$ at $E$ and that of $AD$ and $BC$ at $F$. Let $EP$ and $FQ$ be tangents to this circle with points of tangency $P$ and $Q$ respectively. Suppose…

geometry triangles problem-solving power-of-the-point

asked Nov 01 '20 at 17:09

Anonymous

4,310

votes

4 answers

A problem concerning a parallelogram and a circle

Sorry for the ambiguous title. If you can phrase it better, feel free to edit. "A parallelogram $ABCD$ has sides $AB = 16$ and $AD = 20$. A circle, which passes through the point $C$, touches the sides $AB$ and $AD$, and passes through sides $BC$…

geometry euclidean-geometry triangles circles power-of-the-point

asked Jun 28 '20 at 15:54

Ebrin

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1 answer

Geometry problem I am having trouble to solve

Prove that $AC = \sqrt{ab}$ $a$ is $AB$; $b$ is $CD$; the dot is the origin of the circle. ABCD is a trapezoid, meaning AB || DC. My attempt at solving: According to this rule, $$MA^2 = MB \cdot MC$$ I can apply this rule and say that $DA^2 =…

geometry euclidean-geometry power-of-the-point

asked Jun 16 '20 at 20:01

Witherik

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3 answers

Circle Geometry Colinear Points Problem

I was digging through some old questions I had from high school and I came across this circle geometry problem. . There were no solutions unfortunately. How can this be proven? Here are a few things I've tried: Pascal's theorem Proof that $\angle…

geometry euclidean-geometry circles power-of-the-point

asked Apr 10 '20 at 08:05

LogicAndTruth

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3 answers

Circles tangential to the real line

For any two distinct points $a,b$ in the upper half plane, prove that there is a circle passing through $a$ and $b$ that is tangential to the extended real line. Seems obvious, but how would you go about proving this?

geometry euclidean-geometry circles power-of-the-point

asked Oct 25 '17 at 19:20

F Murray

votes

1 answer

Prove that $\frac{AP}{PC} = 2\cdot \frac{AB}{BC}$ for intersecting chords and tangents

Let point $A$ be outside of circle $O$, where a secant through $A$ intersects $O$ at $ B $ and $ C $ where $ B $ is between $ A $ and $C$. The two tangents from $A$ touch $O$ at $S$ and $T$. Let $AC$ intersect $ST$ at $P$. Prove that …

geometry circles power-of-the-point

asked Apr 29 '25 at 23:06

Tim Macdonald

votes

1 answer

Power of a point theorem proof question

I’m trying to follow the following proof: Wikipedia link It starts with: Let $P:\vec p$ be a point, $c: \vec x^2-r^2=0$ a circle with the origin as its center. And then shows a picture of a circle. How is this equation $c: \vec x^2-r^2=0$ a circle?…

geometry vectors circles power-of-the-point

asked Sep 20 '24 at 04:09

notaorb

votes

1 answer

Given the feet of the altitudes of $\triangle ABC,$ point $R$ and the midpoint $P$ of $\overline{AB}$, prove$ |RA|\cdot|RB|=|RP|\cdot|RN|$

Let $K,M,N$ be feet of the altitudes of $\triangle ABC$ from the vertices $A,B,C$ respectively, $P$ be the midpoint of the edge $\overline{AB}$ and $R$ be the intersection point of the lines $AB$ and $KM$.…

geometry euclidean-geometry circles power-of-the-point

asked Jun 25 '20 at 19:51

Matcha Latte

4,665
4
16
49

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2 answers

Circles and Tangents property

In the following figure $AB=5$ and $BC=4$. I have to find radius of the sector. Somebody help me with this question

geometry euclidean-geometry power-of-the-point

asked Dec 25 '19 at 19:35

Kshitij Singh

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