Category Archives: Geometry

MATHCOUNTS Exercises – 2

Posted on December 22, 2019 by kevin

MASSCOUNTS 2016-2017 – 232 In Circle A, shown here, $\overleftrightarrow{BD}$ is tangent to the circle at B, and major $\overparen{BC\ }$ has measure $230^\circ$. What is $m\angle{CBD}$? Click here for the solution. Solution Draw lines $\overline{AB}$ and $\overline{AC}$ $$\begin{align} &\because … Continue reading →

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MATHCOUNTS Exercises – 1

Posted on December 21, 2019 by kevin

MATHCOUNTS 2016-2017 – 231 Regular nonagon ABCDEFGHI is inscribed in a circle, as shown. What is $m\angle{AHC}$? Click here to show the solutions. Solution 1 Draw line $\overline{AC}$. $$ \begin{align} &\because\ ABCDEFGHI\ \text{is a regular nonagon} \\ &\therefore \triangle{ABC}, \triangle{AIH}, … Continue reading →

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Circles in a Square – Part 10

Posted on December 17, 2019 by kevin

In this 10th post of this series, consider the following figure with two co-tangent semi-circles drawn, centered at point $E$ on $\overline{AB}$ and $F$ on $\overline{BC}$. A full circle centered at point G, tangent with both semi-circles, line $\overline{AD}$ and $\overline{CD}$ in a unit square $ABCD$. Find … Continue reading →

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Circles in a Square – Part 9

Posted on December 9, 2019 by kevin

Continue the topic of the previous post in this series, if we construct a semi-circle first, then a quarter circle, and finally a full circle, as the following, what the radius of the full circle? The problem is harder than the … Continue reading →

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Problem of the Day – December 5, 2019

Posted on December 5, 2019 by kevin

In $\triangle ABC$, $\dfrac{AD}{BD} = \dfrac{1}{2}$, $\dfrac{BE}{EC} = \dfrac{1}{3}$, and $\dfrac{AF}{CF} = \dfrac{3}{2}$. What is the ratio of the area of $\triangle GHI$ to $\triangle ABC$?

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