**These problems are my submission to Xuming's Geometry group.These problems are taken from various sources.**

- Q1 In a triangle ABC altitudes are drawn from the vertices A,B & C at points L,M & N respectively.

Prove that : \(\frac{AN}{NB}\) * \(\frac{BL}{LC}\) * \(\frac{CM}{MA}\)=1

-Q2 Prove that in any triangle 3 times the sum of squares of the length of sides of triangle is equal to 4 times the sum of squares of the length of medians of the triangle.

-Q3 In a parallelogram ABCD a line segment is drawn from D which meets AB at at F(produced).This line segment DF passes through BC at E(which is the mid point of BC) and through diagonal AC at O

Prove that:OA=\(2\times OC\)

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We have \(\Delta OAD\) similar to \(\Delta OCE\) by the \(AAA\) criterion. (\(\angle DOA = \angle COE\), \(\angle ODA = \angle OEC\), \(\angle OAD = \angle OCE\)). Furthermore, \(2CE = AD\), which implies that \(\Delta OAD\) is an enlargenment of \(\Delta OCE\) by a factor of 2. This implies that \(OA = 2OC\) since they are corresponding sides on the triangles. Therefore, \(OA = 2 \times OC\). – Sharky Kesa · 1 year ago

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Nicely explained– Ambuj Shrivastava · 1 year agoLog in to reply

Q2) From Appolonius' theorem, We have

\(AB^2 + AC^2 = 2(AD^2+ \dfrac {BD^2}{4}) \\ AB^2 + BC^2 = 2(BE^2+ \dfrac {AD^2}{4}) \\ AC^2 + BC^2 = 2(CF^2+ \dfrac {AB^2}{4})\)

Adding all these, and transposing (I'm sorry for posting the incomplete solution, It's night here.)

\(3 (AB^2+BC^2+CA^2)= 4(AD^2+BE^2+CF^2)\) – Mehul Arora · 1 year ago

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Q1) In any triangle, a line drawn from a vertex to the opposite side is called a cevian. Thus, AL, BM and CN are cevians.

It directly follows from Ceva's theorem that \(\dfrac {AN}{BL} \times \dfrac {BL}{BL} \times \dfrac {CM}{MA}=1\) – Mehul Arora · 1 year ago

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Its complete proof would be as followstriangle ALB similar to triangle CNB by AA criteria. .. \(\Rightarrow\) \(\frac {LB}{NB}\) = \(\frac {AL}{CN}\) \(\rightarrow\) 1triangle ACL similar to triangle BCM by AA criteria. \(\Rightarrow\) \(\frac {CM}{CL}\) = \(\frac {BM}{AL}\) \(\rightarrow\) 2triangle BMA similar to triangle CNA by AA criteria.. \(\Rightarrow\) \(\frac {NA}{MA}\) = \(\frac {CN}{BM}\) \(\rightarrow\) 3Now, On multiplying 1,2 & 3.and rearranging we get

\(\frac{AN}{NB}\) * \(\frac{BL}{LC}\) * \(\frac{CM}{MA}\)=1

As required. – Ambuj Shrivastava · 1 year ago

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– Mehul Arora · 1 year ago

That's how we prove Ceva's theorem. It's the same thing, although you proved Ceva's theorem. whereas I directly used itLog in to reply

– Ambuj Shrivastava · 1 year ago

Sorry I didn't know about that theorem so i thought it to be a good question but still the proof might be useful.Log in to reply

– Mehul Arora · 1 year ago

Yeah, It's okay. :)Log in to reply

– Ambuj Shrivastava · 1 year ago

Am i allowed to add more questions to the set because i recently found one.Log in to reply

– Mehul Arora · 1 year ago

Well, maybe you can add it as a different note for the benefit of everyone.Log in to reply

– Ambuj Shrivastava · 1 year ago

Okay thank you.I have a doubt regarding cevea theorem can you please explain that how can we prove it in case of an ordinary cevian that is when its not an altitude, angle bisector or median.Log in to reply

– Mehul Arora · 1 year ago

Ceva's theorem states that \(\dfrac {AN}{BL} \times \dfrac {BL}{BL} \times \dfrac {CM}{MA}=1\) is true for any three cevians of a triangle, not only angle bisectors, or perpendiculars, or MediansLog in to reply

– Ambuj Shrivastava · 1 year ago

Yeah but I was thinking about its proof!!Log in to reply

– Sharky Kesa · 1 year ago

You typoed the theorem.Log in to reply

– Mehul Arora · 1 year ago

Please change it, my lord.Log in to reply