June 2022

 

2022/06/07

a,b,c>0,a32+b32+c32=1, prove:

(1) abc19 (2) ab+c+bc+a+ca+b12abc

prove: (1)G3=x1x2x33,A3=x1+x2+x33,GnAna32+b32+c32=1a32+b32+c323=13a32b32c323=(abc)1213abc19(2)ab+c+bc+a+ca+ba2bc+b2ca+c2abab+c+bc+a+ca+ba32+b32+c322abc=12abc


2022/06/21

In ABC, D is a point on BC so that DAC=60,ACB=40,AC=BD, find ABC.

image-20220621195654015

Solve:

image-20220621205710051 Easy to see ADC=1806040=80Extend AD to E so that ACE is equilateral.AE=AC=CE,DCE=20Make F on BC so that CE=EFCE=CF=AC=BD,CEF=CFE=80=ADC=FDEBF+DF=CD+DF,FED=20,EF=DE,BFE=CDEBF=CDBEFCEDBE=CE=AEE is the circumcenter of ABCABC=AEC2=30


2022/06/24

In quadrileteral ABCD, BC=CD,ABD=30,ACD=38,BAC=22, find CAD.

image-20220624194446989

Solve:

image-20220624194334651 CDB=BAC+ABDACD=14=CBDBCD=18021438=114Make F on AB so that CF=BC=CDCFB=CBF=30+14=44FCB=180244=92ACF=11492=22=BACDCF=38+22=60,AF=CFCDF is equilateral triangleCFD=60,CF=DF=AFF is circumcenter of ACDCAD=CFD2=602=30


2022/06/25

In quadrilateral ABCD, CAD=30,CAB=20,ABD=30,CBD=10, find BDC.

image-20220625041914354

Solve:

image-20220625041826709 Make F on AB so that BC=CF,Make D on AD so that AF=FD,Connect CD,BDCFB=CBF=30+10=40,ADF=DAF=30+20=50ACF=CFBCAF=4020=20=CAF,AFD=80AF=FD=CF=BC,CFD=180AFDCFB=1808040=60CFD is equilateral triangleCD=CF=CBCis circumcenter of BFDFBD=FCD2=602=30=FBDD=DBDC=BDC=CBD=CBD=10