## Search found 155 matches

- Sun Nov 22, 2015 1:18 am
- Forum: Secondary Level
- Topic: geometry
- Replies:
**3** - Views:
**8068**

### Re: geometry

$BD$ will be a tangent $\Leftrightarrow OB\perp BD \Leftrightarrow OB||AC \Leftrightarrow \angle ABO=\angle BAC \Leftrightarrow \angle BAO=\angle BAC$

- Wed May 20, 2015 12:51 am
- Forum: Geometry
- Topic: Geometric Inequality
- Replies:
**1** - Views:
**1809**

### Re: Geometric Inequality

Any hints? I'm stuck...

- Mon May 04, 2015 8:39 pm
- Forum: Geometry
- Topic: Inscribed-Quad in an Excribed-Quad
- Replies:
**5** - Views:
**3173**

### Re: Inscribed-Quad in an Excribed-Quad

Extremely sorry for the typo...

- Sun May 03, 2015 2:35 pm
- Forum: Geometry
- Topic: Inscribed-Quad in an Excribed-Quad
- Replies:
**5** - Views:
**3173**

### Inscribed-Quad in an Excribed-Quad

The quadrilateral $ABCD$ is excribed around a circle with centre $I$. Prove that, the projections of $B$ and $D$ on $IA$, $IC$ lie on a circle.

*Sharygin Geometry Olympiad, Russia*- Sun May 03, 2015 1:43 am
- Forum: Geometry
- Topic: triangular inequality [sides and area]
- Replies:
**3** - Views:
**2311**

### Re: triangular inequality [sides and area]

This was also set as Problem No. 2 in the 1961 IMO.

- Wed Apr 29, 2015 9:39 pm
- Forum: Algebra
- Topic: Inequality (sin, r and s)
- Replies:
**2** - Views:
**2025**

### Inequality (sin, r and s)

Prove the inequality,

\[\frac{1}{\sqrt{2\sin A}}+\frac{1}{\sqrt{2\sin B}}+\frac{1}{\sqrt{2\sin C}}\leq \sqrt{\frac{s}{r}}\]

where, $s$ and $r$ are the semi-perimeter and inradius of $\triangle ABC$.

\[\frac{1}{\sqrt{2\sin A}}+\frac{1}{\sqrt{2\sin B}}+\frac{1}{\sqrt{2\sin C}}\leq \sqrt{\frac{s}{r}}\]

where, $s$ and $r$ are the semi-perimeter and inradius of $\triangle ABC$.

*Sharygin Geometry Olympiad, Russia*### Re: cool geo

Corrected.

### Re: cool geo

$AEHF$ is cyclic. $\therefore P \in \odot AEF$ iff $\angle APH= \angle AFH =90^{\circ}$. Suppose, $AP\cap\odot ABC=G$. Since, $G$ is the mid-point of arc $\widehat{BC}$, $M$ is the mid-point of $OG$. Now, $AN||OM$ and $AN=OM$ $\Rightarrow AOMN$ is a prallelogram $\Rightarrow AO||MN$ $\Rightarrow \tr...

- Tue Feb 24, 2015 10:15 pm
- Forum: Higher Secondary Level
- Topic: Vectors around Regular Polygon
- Replies:
**4** - Views:
**4344**

### Re: Vectors around Regular Polygon

Hmm, Nayel Vai... Complex Numbers definitely give a straightforward solution. But, Mahi's approach was very neat. :) Solution using complex numbers: Let, the polygon be inscribed in the unit-circle, and, $OP_1$ be the real axis. Then, \[\overrightarrow{OP_i}=\cos\frac{2\pi}{n}(i-1)+i\cdot\sin\frac{2...

- Sat Feb 21, 2015 11:25 pm
- Forum: Higher Secondary Level
- Topic: Vectors around Regular Polygon
- Replies:
**4** - Views:
**4344**

### Vectors around Regular Polygon

Let, $ P_1P_2P_3\ldots P_n$ be a regular polygon whose circumcentre is $O$. Prove that,

\[\sum_{i=1}^n \overrightarrow{OP_i}=0\]

\[\sum_{i=1}^n \overrightarrow{OP_i}=0\]