Let \(ABCD\) be a convex quadrilateral with \(\angle DAB = \angle BDC = 90^\circ\). Let the incircles of triangles \(ABD\) and \(BCD\) touch \(BD\) at \(P\) and \(Q\), respectively, with \(P\) lying in between \(B\) and \(Q\). If \(AD = 999\) and \(PQ = 200\) then what is the sum of the radii of the incircles of triangles \(ABD\) and \(BDC\)?

This note is part of the set Pre-RMO 2014

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestQ12

From the figure we can see that \(SD = PD = PQ + QD = r_1 + 200\) and \(AD = AS + SD =r_ 2 + SD = 999\).

\(r_ 2 + r_1 + 200 = 999\)

\(r_1+ r_2 = \boxed{799}\)

Log in to reply

799

Log in to reply

Let \(O_1\) and \(O_2\) be the incenters of \( \bigtriangleup DAB\) and \( \bigtriangleup CDB\) respectively. Draw perpendicular from \(O_1\) to sides \(AD, DB\) and \(AB\) at points \(S, P\) and \(E\) respectively. Draw perpendicular from \(O_2\) to sides \(CD\) and \(DB\) at points \(F\) and \(Q\) respectively. Let \(R\) and \(r\) be the inradii of \( \bigtriangleup DAB\) and \( \bigtriangleup CDB\) respectively.

\( SO_1EA\) and \(FO_2QD\) are squares with side \(r\) and \(R\) respectively.

\(DS=AD-R=999-r\)

\(DP=DQ+PQ=R+200\)

Since \(DP\) and \(DS\) are tangents from the same point they are equal in length.

\( \Rightarrow 999-r=R+200\)

\( \Rightarrow R+r=999-200= \boxed{799}\)

Log in to reply

Why 799???

Log in to reply

499.5 sqrt 2

Log in to reply

799

Log in to reply