## regional mo 2015

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kh ibrahim
Posts: 17
Joined: Mon May 09, 2016 11:18 am

### regional mo 2015

this is a problem of regional math olympiad 2015
Attachments rmo 2016
rps20161210_124814.jpg (57.18 KiB) Viewed 1746 times

kh ibrahim
Posts: 17
Joined: Mon May 09, 2016 11:18 am

### Re: regional mo 2015

Disclose the first approach

asif e elahi
Posts: 183
Joined: Mon Aug 05, 2013 12:36 pm

### Re: regional mo 2015

Use the Power of Point theorem to prove that $PA\times PD=PE^2=PB\times PC$

Absur Khan Siam
Posts: 65
Joined: Tue Dec 08, 2015 4:25 pm
Location: Bashaboo , Dhaka

### Re: regional mo 2015

asif e elahi wrote:Use the Power of Point theorem to prove that $PA\times PD=PE^2=PB\times PC$
$PA \times PD = 12 \times (PA + AB + BC + CD) = 12 \times (12 + AB + BC + 2AB)$
$= 36AB + 12BC + 144...(i)$
$PB \times PC = (PA+AB) \times (PA + AB + BC) = (12+AB) \times (12 + AB + BC)$
$= AB^2 + 24AB + 12AB + AB \times BC ...(ii)$
$(i) = (ii) \rightarrow 36AB + 12BC + 144 = AB^2 + 24AB + 12AB + AB \times BC$
$\rightarrow 12AB = AB(AB+BC)$
Thus,$AB+BC = 12$
And we get , $PC = PA + AB + BC = 24$ "(To Ptolemy I) There is no 'royal road' to geometry." - Euclid