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Question Number 37220 by tawa tawa last updated on 10/Jun/18

Answered by ajfour last updated on 01/Jul/18

let x axis be DEF sum of perpendiculars on tangent (x axis now)=y_A +y_B +y_C =3y_G y_G being y coordinate of centroid of △ABC (same as y coordinate of circumcentre) =R Length of median=R+(R/2)=((3R)/2) hence y_A +y_B +y_C =3y_G ________________________ We can see that centroid and circumcentre are the same for an equilateral triangle; y_G = R as circle touches x-axis. Hence y_A +y_B +y_C = 3R which is twice the length of a median ( = R+Rsin 30°) = ((3R)/2) ________________________ =2(length of median).

$${let}\:{x}\:{axis}\:{be}\:{DEF} \\ $$$${sum}\:{of}\:{perpendiculars}\:{on}\:{tangent} \\ $$$$\left({x}\:{axis}\:{now}\right)={y}_{{A}} +{y}_{{B}} +{y}_{{C}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{3}{y}_{{G}} \\ $$$${y}_{{G}} \:{being}\:{y}\:{coordinate}\:{of}\:{centroid} \\ $$$${of}\:\bigtriangleup{ABC}\:\left({same}\:{as}\:{y}\:{coordinate}\right. \\ $$$$\left.{of}\:{circumcentre}\right)\:={R} \\ $$$${Length}\:{of}\:{median}={R}+\frac{{R}}{\mathrm{2}}=\frac{\mathrm{3}{R}}{\mathrm{2}} \\ $$$${hence}\:\:{y}_{{A}} +{y}_{{B}} +{y}_{{C}} =\mathrm{3}{y}_{{G}} \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$${We}\:{can}\:{see}\:{that}\:{centroid}\:{and} \\ $$$${circumcentre}\:{are}\:{the}\:{same}\:{for} \\ $$$${an}\:{equilateral}\:{triangle};\:{y}_{{G}} \:=\:{R} \\ $$$${as}\:{circle}\:{touches}\:{x}-{axis}. \\ $$$$\:\:{Hence}\:\:\:{y}_{{A}} +{y}_{{B}} +{y}_{{C}} \:=\:\mathrm{3}{R} \\ $$$${which}\:{is}\:{twice}\:{the}\:{length}\:{of}\:{a} \\ $$$${median}\:\left(\:=\:{R}+{R}\mathrm{sin}\:\mathrm{30}°\right)\:=\:\frac{\mathrm{3}{R}}{\mathrm{2}} \\ $$$$\:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2}\left({length}\:{of}\:{median}\right). \\ $$

Commented by tawa tawa last updated on 11/Jun/18

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Commented by tawa tawa last updated on 28/Jun/18

please sir, can you produce more explanation on the solution for me to understand better. Thanks you sir and God bless you.

$$\mathrm{please}\:\mathrm{sir},\:\mathrm{can}\:\mathrm{you}\:\mathrm{produce}\:\mathrm{more}\:\mathrm{explanation}\:\mathrm{on}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{for}\:\mathrm{me} \\ $$$$\mathrm{to}\:\mathrm{understand}\:\mathrm{better}.\:\mathrm{Thanks}\:\mathrm{you}\:\mathrm{sir}\:\mathrm{and}\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$

Commented by tawa tawa last updated on 01/Jul/18

?

$$? \\ $$

Commented by tawa tawa last updated on 01/Jul/18

wow, i really appreciate sir. God bless you sir

$$\mathrm{wow},\:\mathrm{i}\:\mathrm{really}\:\mathrm{appreciate}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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