Sine in A.P.
Geometry
Level
3
If in a triangle PQR , are in A.P. then
Also try : Cot in A.P.
The altitudes are in A.P.
The medians are in A.P.
The medians are in G.P.
The altitudes are in H.P
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1 solution
Let sides of the triangle be a,b,c.Then sinP=a/2Z where Z is circumradius,similarly sinQ=b/2Z and sinR=c/2Z. As, sinP,sinQ and sinR are in AP, sinP +sinR =2sinQ. Substituting sinP=a/2Z,sinQ=b/2Z and sin R=c/2Z, We get a+c=2b. Now let area of triangle be X.let perpendiculars onto sides a,b,c be p1 ,p2,p3 respectively from the opposite vertices. Area of triangle=X=p1a/2=p2b/2=p3c/2. Then a=2X/p1,b=2X/p2 and c=2X/p3. As a+c=2b,substituting the values,of a,b and c, We get 1/p1 +1/p3=2/p2. Therefore they are in HP .