Category Archives: Geometry

2022 AMC 10B Problem 20

Posted on November 18, 2022 by kevin

Let $ABCD$ be a rhombus with $\angle{ADC}=46^{\circ}$, and let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle{BFC}$? (A) 110      (B) 111      (C) 112      (D) 113      (E) 114      Solution Draw the … Continue reading →

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MathCounts Geometry Exercise – 2

Posted on November 4, 2022 by kevin

A B C D E P ________ The area of $\triangle{ABC}$ is $1$. $BD:DC=2:1$, and $E$ is the midpoint of $AC$. $AD$ intersects $BE$ at point $P$. Find the area of quadrilateral $PDCE$. ________ Rectangle $ABCD$ is divided into 4 … Continue reading →

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MathCounts Geometry Exercise – 1

Posted on October 28, 2022 by kevin

25 20 30 ________ One rectangle is divided into four smaller rectangles. The areas of three smaller rectangles are $25\ cm^2$, $20\ cm^2$, and $30\ cm^2$ respectively, as shown in the diagram. Find the area of the shaded region (unit: … Continue reading →

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Distance between the incenter and circumcenter

Posted on October 7, 2022 by kevin

The distance between the incenter and circumcenter of a triangle, is calculated by Euler’s theorem in geometry: $$d=\sqrt{R(R-2r)}$$ which also implies $$R\ge2r$$ For a Bicentric quadrilateral, the distance between the incenter and circumcenter can be calculated by Fuss’s theorem or … Continue reading →

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Geometry – Pythagoras’ Theorem

Posted on May 6, 2022 by kevin

In $\triangle{ABC}$, $AM$ is the median on the side $BC$. Prove that $AB^2+AC^2=2(AM^2 + BM^2)$ For $\triangle{ABC}$, $O$ is an inner point, and $D$, $E$, $F$ are on $BC$, $CA$, $AB$ respectively, such that $OD\perp BC$, $OE\perp CA$, and $OF\perp … Continue reading →

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