If D,E,F are the points of contact of the inscribed circle with the sides BC,CA,AB of a triangle , shew that if the squares of AD,BE,CF are in arithmetic progression, then the sides of the triangle are in harmonic progression.

Asked by Dev | 21st Jun, 2016, 11:34: PM

Expert Answer:

 
begin mathsize 12px style In space the space figure space above comma space the space incircle space of space the space triangle space ABC space is space drawn.
Using space tangent space properties comma space Bf equals BD equals straight x comma
CD equals CE equals straight y
AF equals AE equals straight z
If space apostrophe straight s apostrophe space is space the space semi minus perimeter space of space the space triangle space ABC comma space then
2 straight s equals 2 open parentheses straight x plus straight y plus straight z close parentheses rightwards double arrow straight x plus straight y plus straight z equals straight s
straight x plus straight y equals straight a
straight y plus straight z equals straight b
straight z plus straight x equals straight c
Using space these space we space get space BD equals BF equals straight x equals straight s minus straight b
In space triangle space ABD comma space applying space cosine space rule comma space we space get
AD squared equals straight c squared plus open parentheses straight s minus straight b close parentheses squared minus 2 straight c open parentheses straight s minus straight b close parentheses cosB
rightwards double arrow AD squared equals open parentheses straight c minus open parentheses straight s minus straight b close parentheses close parentheses squared plus 2 straight c open parentheses straight s minus straight b close parentheses open parentheses 1 minus cosB close parentheses
rightwards double arrow AD squared equals open parentheses straight c minus open parentheses straight s minus straight b close parentheses close parentheses squared plus 2 straight c open parentheses straight s minus straight b close parentheses cross times 2 sin squared straight B over 2
Using space the space sin space half space angle space formula comma space sin squared straight B over 2 equals fraction numerator open parentheses straight s minus straight c close parentheses open parentheses straight s minus straight a close parentheses over denominator ac end fraction
So space we space get comma
AD squared equals open parentheses straight c minus open parentheses straight s minus straight b close parentheses close parentheses squared plus 2 straight c open parentheses straight s minus straight b close parentheses cross times 2 cross times fraction numerator open parentheses straight s minus straight c close parentheses open parentheses straight s minus straight a close parentheses over denominator ac end fraction
rightwards double arrow AD squared equals open parentheses straight c minus open parentheses straight s minus straight b close parentheses close parentheses squared plus fraction numerator 4 triangle squared over denominator as end fraction space space space open parentheses multiplying space and space dividing space by space apostrophe straight s apostrophe space and space using space heron apostrophe straight s space formula close parentheses
rightwards double arrow AD squared equals open parentheses straight b plus straight c minus straight s close parentheses squared plus fraction numerator 4 straight r triangle over denominator straight a end fraction space left parenthesis using space straight r equals triangle over straight s right parenthesis
rightwards double arrow AD squared equals open parentheses straight s minus straight a close parentheses squared plus fraction numerator 4 straight r triangle over denominator straight a end fraction space left parenthesis using space straight r equals triangle over straight s right parenthesis

Similarly space we space get space
BE squared equals open parentheses straight s minus straight b close parentheses squared plus fraction numerator 4 straight r triangle over denominator straight b end fraction
and comma space CF equals open parentheses straight s minus straight c close parentheses squared plus fraction numerator 4 straight r triangle over denominator straight c end fraction

Since comma space AD squared comma space BE squared space and space CF squared space are space in space straight A. straight P.
BE squared minus AD squared equals CF squared minus BE squared
rightwards double arrow open parentheses straight s minus straight b close parentheses squared minus open parentheses straight s minus straight a close parentheses squared plus 4 straight r triangle open parentheses 1 over straight b minus 1 over straight a close parentheses equals open parentheses straight s minus straight c close parentheses squared minus open parentheses straight s minus straight b close parentheses squared plus 4 straight r triangle open parentheses 1 over straight c minus 1 over straight b close parentheses
rightwards double arrow open parentheses straight a minus straight b close parentheses open parentheses 2 straight s minus straight a minus straight b close parentheses plus 4 straight r triangle open parentheses 1 over straight b minus 1 over straight a close parentheses equals open parentheses straight b minus straight c close parentheses open parentheses 2 straight s minus straight b minus straight c close parentheses plus 4 straight r triangle open parentheses 1 over straight c minus 1 over straight b close parentheses
rightwards double arrow open parentheses straight a minus straight b close parentheses straight c plus 4 straight r triangle open parentheses 1 over straight b minus 1 over straight a close parentheses equals open parentheses straight b minus straight c close parentheses straight a plus 4 straight r triangle open parentheses 1 over straight c minus 1 over straight b close parentheses
rightwards double arrow 2 ac minus straight b open parentheses straight a plus straight c close parentheses equals 4 straight r triangle open parentheses 1 over straight c plus 1 over straight a minus 2 over straight b close parentheses
rightwards double arrow 2 ac minus straight b open parentheses straight a plus straight c close parentheses equals 4 straight r triangle open parentheses fraction numerator open parentheses straight a plus straight c close parentheses straight b minus 2 ac over denominator abc end fraction close parentheses
The space above space equation space is space satisfied space when space
2 ac equals straight b open parentheses straight a plus straight c close parentheses
rightwards double arrow 2 over straight b equals 1 over straight c plus 1 over straight a space open parentheses dividing space both space sides space by space apostrophe abc apostrophe close parentheses
Which space is space the space required space condition space for space straight a comma space straight b comma space straight c space to space be space in space straight H. straight P.
Hence comma space proved end style

Answered by satyajit samal | 23rd Jun, 2016, 12:29: PM