## [OGC1] Online Geometry Camp: Day 4

Discussion on Bangladesh National Math Camp
nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### [OGC1] Online Geometry Camp: Day 4

"Everything should be made as simple as possible, but not simpler." - Albert Einstein

Tusher Chakraborty
Posts: 27
Joined: Wed Oct 24, 2012 10:16 pm
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### Re: [OGC1] Online Geometry Camp: Day 4

আজকে ক্যাম্পে আলোচনার শেষ দিন। কালকে Day-1 ও Day-2 এর ওপর পরীক্ষা। সকাল ৮ টায় প্রশ্ন আপলোড দেওয়া হবে।
আজকের টপিকঃ
বৃত্ত সংক্রান্ত উপপাদ্য- ৪.১ হতে ৪.৮(মাধ্যমিক উচ্চতর জ্যামিতি)
"Your present circumstances don't determine where you can go; they merely determine where you start." -Nido Qubein

Tusher Chakraborty
Posts: 27
Joined: Wed Oct 24, 2012 10:16 pm
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### Re: [OGC1] Online Geometry Camp: Day 4

একদম সহজ একটা প্রবলেম দিয়ে শুরু করি। এটা এবারের ঢাকা-১ এ ছিল।
Attachments Problem1.PNG (87.51 KiB) Viewed 5118 times
"Your present circumstances don't determine where you can go; they merely determine where you start." -Nido Qubein

Nahraf
Posts: 8
Joined: Sun Aug 25, 2013 2:28 pm
Location: Gazipur

### Re: [OGC1] Online Geometry Camp: Day 4

ভাইয়া, এটাচমেন্টটা শো করছে না৷ একটু লেখে দিবেন? অথবা pdf?
"And that there is not for man except that for which he strives."
Al Quran, 53:39

Mursalin
Posts: 17
Joined: Thu Aug 22, 2013 9:11 pm

### Re: [OGC1] Online Geometry Camp: Day 4

Here are a couple of problems for you to practise. These problems are from older sets in Brilliant.org.

1)A circle $\Gamma$ cuts the sides of a equilateral triangle $ABC$ at $6$ distinct points. Specifically, $\Gamma$ intersects $AB$ at points $D$ and $E$ such that $A$,$D$,$E$,$B$ lie in order. $\Gamma$ intersects $BC$ at points $F$ and $G$ such that $B$,$F$,$G$,$C$ lie in order. $\Gamma$ intersects $CA$ at points $H$ and $I$ such that $C$,$H$,$I$,$A$ lie in order. If $|AD|=3$, $|DE|=39$, $|EB|=6$ and $|FG|=21$, what is the value of $|HI|^2$?

Hint: Try using the intersecting secants theorem. See the original problem here.

2)Circles $\Gamma_1$ and $\Gamma_2$ have centers $X$ and $Y$ respectively. They intersect at points $A$ and $B$, such that angle $XAY$ is obtuse. The line $AX$ intersects $\Gamma_2$ again at $P$, and the line $AY$ intersects $\Gamma_1$ again at $Q$. Lines $PQ$ and $XY$ intersect at $G$, such that $Q$ lies on line segment $GP$. If $GQ=255$, $GP=266$ and $GX=190$, what is the length of $XY$?

Hint: Power-of-a-point or noticing cyclic quads could help. The original problem is here.

3)$ABC$ is a triangle with $AC=139$ and $BC=178$. Points $D$ and $E$ are the midpoints of $BC$ and $AC$ respectively. Given that $AD$ and $BE$ are perpendicular to each other, what is the length of $AB$?

Hint: The centroid of a triangle trisects its medians. The original problem.

4)$ABCD$ is a trapezoid(US) or Trapezium(UK) with parallel sides $AB$ and $CD$. $\Gamma$ is an inscribed circle of $ABCD$, and tangential to sides $AB$,$BC$,$CD$ and $AD$ at the points $E$,$F$,$G$ and $H$ respectively. If $AE=2$,$BE=3$, and the radius of $\Gamma$ is $12$, what is the length of $CD$?

Hint: There are multiple ways to solve this problem. Are you familiar with Pitot's Theorem? See the original problem here.

Have fun solving these problems!
"Be boring. It's the only way get work done."

- Austin Kleon

nowshin
Posts: 6
Joined: Sun Jan 16, 2011 9:30 pm

### Re: [OGC1] Online Geometry Camp: Day 4

বৃত্তদ্বয়ের মাঝে যে ছায়া আছে তার ক্ষেত্রফল $\dfrac{\pi}{2} -1$ এবং উপরের ছায়াযুক্ত অংশের ক্ষেত্রফলও $\dfrac{\pi}{2} -1$. অতএব এদের পার্থক্য $0$. সুতরাং $a=0$. Tusher Chakraborty
Posts: 27
Joined: Wed Oct 24, 2012 10:16 pm
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### Re: [OGC1] Online Geometry Camp: Day 4

nowshin wrote:বৃত্তদ্বয়ের মাঝে যে ছায়া আছে তার ক্ষেত্রফল $\dfrac{\pi}{2} -1$ এবং উপরের ছায়াযুক্ত অংশের ক্ষেত্রফলও $\dfrac{\pi}{2} -1$. অতএব এদের পার্থক্য $0$. সুতরাং $a=0$. Details please "Your present circumstances don't determine where you can go; they merely determine where you start." -Nido Qubein

Tusher Chakraborty
Posts: 27
Joined: Wed Oct 24, 2012 10:16 pm
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### Re: [OGC1] Online Geometry Camp: Day 4

Nahraf wrote:ভাইয়া, এটাচমেন্টটা শো করছে না৷ একটু লেখে দিবেন? অথবা pdf?
Image save করে নিয়ে দেখো।
"Your present circumstances don't determine where you can go; they merely determine where you start." -Nido Qubein

Neblina
Posts: 18
Joined: Sun Feb 06, 2011 8:38 pm

### Re: [OGC1] Online Geometry Camp: Day 4

Area of large circle is $4\pi$
Let the area of intersection between the small circles be x
Area of one small circle is pi.
So total area of four small circles= $$4\pi-4x$$.
Let area of upper shadow part be y
So $$4\pi-(4\pi-4x)=4y$$
$$4\pi-4\pi+4x=4y$$
This gives up x=y
So the difference between the two shaded area is $0$. so $a=0$

Tusher Chakraborty
Posts: 27
Joined: Wed Oct 24, 2012 10:16 pm
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### Re: [OGC1] Online Geometry Camp: Day 4

@Neblina, perfect "Your present circumstances don't determine where you can go; they merely determine where you start." -Nido Qubein