Labelling the Centerpoint of an Arc with a greek Letter

by Klaus-Anton, Mar 31, 2023, 10:32 AM

wloy2022 asks for Diagram Help.

For me it is not really clear what he there calls $\alpha$ . Please excuse me.

But putting a label (as a greek letter) to the centerpoint of an arc there normally you use some kind of bisector functions.

Look here what i ansered there:

$[asy] pair A = (5, 12); pair B = (0, 0); pair C = (5 + 2035/69, 0); pair D = (10, 0); pair myE = (5/2 + 2035/138, -400/23); draw(A--B--C--cycle); draw(A--D); draw(B--myE--C); draw(D--myE); draw(anglemark(B, A, D, 75),red); draw(anglemark(D, A, C, 75)); draw(anglemark(myE, B, D, 75)); label("$A$", A, N,red); label("$B$", B, W,blue); label("$C$", C, E); label("$D$", D, NE); label("$E$", myE, S); label("\boldmath$\alpha$",A,3*dir(A--B,A--myE),red); /* asymptote.pdf, page 33 wrote: pair dir(path p, path q) returns unit(dir(p)+dir(q)). */ label("\boldmath$\gamma$",C,7*dir(C--bisectorpoint(A,C,B))*dir(1)); //bisectorpoint of olympiad.asy //*dir(1) here is thought by me as "optical correction". (It rotates the label by 1 degree.) shipout(bbox(2mm,Fill(white))); [/asy]$

Figure 1

label("\boldmath$\alpha$",A,3*dir(A--B,A--myE),red);
/* asymptote.pdf, page 33 wrote:
pair dir(path p, path q)
returns unit(dir(p)+dir(q)).
*/
 
label("\boldmath$\gamma$",C,7*dir(C--bisectorpoint(A,C,B))*dir(1)); //bisectorpoint of olympiad.asy
//*dir(1) here is thought by me as "optical correction". (It rotates the label by 1 degree.)

Code: Excerpt with comments

Now in my next figure there you can see how i try to understand some of the underlying basic ideas of the diagram of wloy2022. Am i right so far?

$[asy] import graph; import geometry; real degreesz1z2(pair z1, pair z2) {return degrees(dir(z1 -- z2));} size(8cm); unitsize(.23cm); pair A = (5, 12); pair B = (0, 0); pair C = (5 + 2035/69, 0); pair D = (10, 0); pair myE = (5/2 + 2035/138, -400/23); pair mpBC=midpoint(B--C); pair O=midpoint(B--C); pair Afoot1=(A.x,0); pair Afoot2=foot(A,C,myE); draw(A--B--C--cycle); draw(A--D); draw(B--myE--C); draw(D--myE); draw(anglemark(B, A, D, 75)); draw(anglemark(D, A, C, 75)); draw(anglemark(myE, B, D, 75)); real degreesBA=degreesz1z2(B,A); real degreesBE=degreesz1z2(B,myE); real degreesCE=degreesz1z2(C,myE); real degreesEC=degreesz1z2(myE,C); real degreesEA=degreesz1z2(myE,A); real degreesOA=degreesz1z2(O,A); real theta=degreesz1z2(O,A); draw(arc(mpBC,2.8,0,degreesz1z2(mpBC,A)),blue); draw(arc(B,3.8,0,degreesBA),blue); //draw(arc(myE,3.8, degreesEC,degreesEA),blue); draw(arc(myE,5.8, 45,45+theta/2),blue); label("$\theta$",mpBC,dir(.5*degreesz1z2(mpBC,A)),blue); label("$\theta/2$",B,dir(.2*degreesz1z2(mpBC,A)),blue); label("$\theta/2$",myE,4.5*dir(myE--C,myE--A),blue); //draw(circle(A,arclength(A--B)));//AB=AD? //draw(circle(A,arclength(A--C)));// AC=AE? //draw(arc(myE,5,degreesEC,degreesEC+90)); draw(A--(A.x,0),red); draw(A--mpBC,red); draw(O--Afoot2,red+Dotted); draw(circumcircle(A,B,C),red); //label("$A$", A, N); label("$A$", A, dir(mpBC--A)); label("$B$", B, W); label("$C$", C, E); label("$D$", D, NE); label("$E$", myE, S); label("$O$",O,dir(O--B,A--O),red); //label("$\alpha$", D, dir(D--B,D--myE));//??? dot(mpBC,red,Fill(white)); perpfactor=.8; //perpendicularmark(myE,dir(myE--B,myE--C),blue); perpendicularmark(myE,N,blue); perpendicularmark(Afoot1,NE,blue); perpendicularmark(Afoot2,dir(degreesEC+45),blue); perpendicularmark(A,dir(A--B,A--C),blue); shipout(bbox(2mm,Fill(white))); [/asy]$

Figure 2

What do we see? What can we make out?

There are two isosceles triangles, what we colorize: 1.) $\Delta~ABD$ and 2.) $\Delta~AEC$ . We also have already added their besectors from $A$ together with the perpendicular marks.

$[asy] import graph; import geometry; real degreesz1z2(pair z1, pair z2) {return degrees(dir(z1 -- z2));} size(8cm); unitsize(.23cm); pair A = (5, 12); pair B = (0, 0); pair C = (5 + 2035/69, 0); pair D = (10, 0); pair myE = (5/2 + 2035/138, -400/23); pair mpBC=midpoint(B--C); pair O=midpoint(B--C); pair Afoot1=(A.x,0); pair Afoot2=foot(A,C,myE); real degreesBA=degreesz1z2(B,A); real degreesBE=degreesz1z2(B,myE); real degreesCE=degreesz1z2(C,myE); real degreesEC=degreesz1z2(myE,C); real degreesEA=degreesz1z2(myE,A); real degreesOA=degreesz1z2(O,A); real theta=degreesz1z2(O,A); fill(A--myE--C--cycle, red+white); //fill(A--B--O--cycle,lightgreen); fill(A--B--D--cycle,mediumred+white); draw(arc(mpBC,2.8,0,degreesz1z2(mpBC,A)),blue); draw(arc(B,3.8,0,degreesBA),blue); //draw(arc(myE,3.8, degreesEC,degreesEA),blue); draw(arc(myE,5.8, 45,45+theta/2),blue); label("$\theta$",mpBC,dir(.5*degreesz1z2(mpBC,A)),blue); label("$\theta/2$",B,dir(.2*degreesz1z2(mpBC,A)),blue); label("$\theta/2$",myE,4.5*dir(myE--C,myE--A),blue); //draw(circle(A,arclength(A--B)));//AB=AD? //draw(circle(A,arclength(A--C)));// AC=AE? //draw(arc(myE,5,degreesEC,degreesEC+90)); draw(A--(A.x,0),red); draw(A--mpBC,red); draw(O--Afoot2,red+Dotted); draw(circumcircle(A,B,C),red); //label("$A$", A, N); label("$A$", A, dir(mpBC--A)); label("$B$", B, W); label("$C$", C, E); label("$D$", D, NE); label("$E$", myE, S); label("$O$",O,dir(O--B,A--O),red); //label("$\alpha$", D, dir(D--B,D--myE));//??? perpfactor=.8; //perpendicularmark(myE,dir(myE--B,myE--C),blue); perpendicularmark(myE,N,blue); perpendicularmark(Afoot1,NE,blue); perpendicularmark(Afoot2,dir(degreesEC+45),blue); perpendicularmark(A,dir(A--B,A--C),blue); draw(A--B--C--cycle); draw(A--D); draw(B--myE--C); draw(D--myE); draw(anglemark(B, A, D, 75)); draw(anglemark(D, A, C, 75)); draw(anglemark(myE, B, D, 75)); dot(A^^B^^C^^D^^myE^^O,red,Fill(white)); shipout(bbox(2mm,Fill(white))); [/asy]$

Figure 3: Two red Isosceles Triangles

And more? Look depper! Ah!!! I see. There is a third one: 3.) $\Delta~ABO$ . Let us colorize it in green.

$[asy] import graph; import geometry; real degreesz1z2(pair z1, pair z2) {return degrees(dir(z1 -- z2));} size(8cm); unitsize(.23cm); pair A = (5, 12); pair B = (0, 0); pair C = (5 + 2035/69, 0); pair D = (10, 0); pair myE = (5/2 + 2035/138, -400/23); pair mpBC=midpoint(B--C); pair O=midpoint(B--C); pair Afoot1=(A.x,0); pair Afoot2=foot(A,C,myE); pair OfootAB=foot(O,A,B); real degreesBA=degreesz1z2(B,A); real degreesBE=degreesz1z2(B,myE); real degreesCE=degreesz1z2(C,myE); real degreesEC=degreesz1z2(myE,C); real degreesEA=degreesz1z2(myE,A); real degreesOA=degreesz1z2(O,A); real theta=degreesz1z2(O,A); //fill(A--myE--C--cycle, red+white); fill(A--B--O--cycle,lightgreen); draw(O--OfootAB,red); //fill(A--B--D--cycle,mediumred+white); draw(arc(mpBC,2.8,0,degreesz1z2(mpBC,A)),blue); draw(arc(B,3.8,0,degreesBA),blue); //draw(arc(myE,3.8, degreesEC,degreesEA),blue); draw(arc(myE,5.8, 45,45+theta/2),blue); label("$\theta$",mpBC,dir(.5*degreesz1z2(mpBC,A)),blue); label("$\theta/2$",B,dir(.2*degreesz1z2(mpBC,A)),blue); label("$\theta/2$",myE,4.5*dir(myE--C,myE--A),blue); //draw(circle(A,arclength(A--B)));//AB=AD? //draw(circle(A,arclength(A--C)));// AC=AE? //draw(arc(myE,5,degreesEC,degreesEC+90)); draw(A--(A.x,0),red); draw(A--mpBC,red); draw(O--Afoot2,red+Dotted); draw(circumcircle(A,B,C),red); //label("$A$", A, N); label("$A$", A, dir(mpBC--A)); label("$B$", B, W); label("$C$", C, E); label("$D$", D, NE); label("$E$", myE, S); label("$O$",O,dir(O--B,A--O),red); //label("$\alpha$", D, dir(D--B,D--myE));//??? perpfactor=.8; //perpendicularmark(myE,dir(myE--B,myE--C),blue); perpendicularmark(myE,N,blue); perpendicularmark(Afoot1,NE,blue); perpendicularmark(Afoot2,dir(degreesEC+45),blue); perpendicularmark(A,dir(A--B,A--C),blue); perpendicularmark(OfootAB,dir(degreesBA-45),blue); draw(A--B--C--cycle); draw(A--D); draw(B--myE--C); draw(D--myE); draw(anglemark(B, A, D, 75)); draw(anglemark(D, A, C, 75)); draw(anglemark(myE, B, D, 75)); dot(A^^B^^C^^D^^myE^^O,red,Fill(white)); shipout(bbox(2mm,Fill(white))); [/asy]$

Figure 4: Green Isosceles Triangle
rrrrrrrrrr $\Delta~ABO$ with Base $AB$

$[asy] import graph; import geometry; real degreesz1z2(pair z1, pair z2) {return degrees(dir(z1 -- z2));} size(8cm); unitsize(.23cm); pair A = (5, 12); pair B = (0, 0); pair C = (5 + 2035/69, 0); pair D = (10, 0); pair myE = (5/2 + 2035/138, -400/23); pair mpBC=midpoint(B--C); pair O=midpoint(B--C); pair Afoot1=(A.x,0); pair Afoot2=foot(A,C,myE); pair OfootAB=foot(O,A,B); pair CfootDE=foot(C,D,myE); real degreesBA=degreesz1z2(B,A); real degreesBE=degreesz1z2(B,myE); real degreesCE=degreesz1z2(C,myE); real degreesEC=degreesz1z2(myE,C); real degreesEA=degreesz1z2(myE,A); real degreesOA=degreesz1z2(O,A); real theta=degreesz1z2(O,A); //fill(A--myE--C--cycle, red+white); fill(A--B--O--cycle,lightgreen); draw(O--OfootAB,red); //fill(A--B--D--cycle,mediumred+white); fill(C--D--myE--cycle, mediumgreen); draw(C--CfootDE,red); draw(arc(mpBC,2.8,0,degreesz1z2(mpBC,A)),blue); draw(arc(B,3.8,0,degreesBA),blue); //draw(arc(myE,3.8, degreesEC,degreesEA),blue); draw(arc(myE,5.8, 45,45+theta/2),blue); label("$\theta$",mpBC,dir(.5*degreesz1z2(mpBC,A)),blue); label("$\theta/2$",B,dir(.2*degreesz1z2(mpBC,A)),blue); label("$\theta/2$",myE,4.5*dir(myE--C,myE--A),blue); //draw(circle(A,arclength(A--B)));//AB=AD? //draw(circle(A,arclength(A--C)));// AC=AE? //draw(arc(myE,5,degreesEC,degreesEC+90)); draw(A--(A.x,0),red); draw(A--mpBC,red); draw(O--Afoot2,red+Dotted); draw(circumcircle(A,B,C),red); //label("$A$", A, N); label("$A$", A, dir(mpBC--A)); label("$B$", B, W); label("$C$", C, E); label("$D$", D, NE); label("$E$", myE, S); label("$O$",O,dir(O--B,A--O),red); //label("$\alpha$", D, dir(D--B,D--myE));//??? perpfactor=.8; //perpendicularmark(myE,dir(myE--B,myE--C),blue); perpendicularmark(myE,N,blue); perpendicularmark(Afoot1,NE,blue); perpendicularmark(Afoot2,dir(degreesEC+45),blue); perpendicularmark(A,dir(A--B,A--C),blue); perpendicularmark(OfootAB,dir(degreesBA-45),blue); perpendicularmark(CfootDE,dir(degreesEA-45),blue); draw(A--B--C--cycle); draw(A--D); draw(B--myE--C); draw(D--myE); draw(anglemark(B, A, D, 75)); draw(anglemark(D, A, C, 75)); draw(anglemark(myE, B, D, 75)); //draw(arc(C,arclength(C--D),180,360-45-90)); dot(A^^B^^C^^D^^myE^^O,red,Fill(white)); shipout(bbox(2mm,Fill(white))); [/asy]$

Figure 5: A further green Isosceles Triangle
$\Delta~CDE$ with Base $DE$

Now finally i recall you Klaus-Anton 2023. Trigonometric Identities in the Circle of Thales (which builds on a post of Phu Nguyen). And you say to me: "Fine! - I remember".

$[asy] import graph; import geometry; import markers; usepackage("amsmath"); //usepackage("mtpro2"); //settings.outformat="pdf"; // output is pdf size(7.4cm, keepAspect=true); defaultpen(fontsize(12pt)); void plot_label(picture pic, pair A, pair B, string s, real l) { pair AB=B-A; AB=AB/length(AB); real alpha=degrees(acos(AB.x)); if ( AB.y < 0 ) alpha = -alpha; pair normal=(-AB.y,AB.x); label(pic,rotate(alpha)*s,.5(A+B)+l*normal,blue); } picture pic1, pic2, pic3; // pic 2 real alpha=20; real r = 1; pair O=(0,0); real alpha1=20; real alpha2=20+90; real alpha3=180+45; real alpha4=180+75; draw(pic3,unitcircle,red+1.2pt); // pic1 real alpha1=0; real alpha2=180; real alpha3=90+20; real alpha4=180+85; draw(pic1,unitcircle,red+1.2pt); pair A=dir(alpha2); pair C=dir(alpha1); pair B=dir(alpha3); pair D=dir(alpha4); draw(pic1,O--A,black+.8pt); draw(pic1,O--C,black+.8pt); draw(pic1,B--D,black+.8pt); draw(pic1,A--B--C--D--cycle,black+.8pt); draw(pic1, O--B,red); //label(pic1,"$O$",(0.02,-0.2),red); label(pic1,"$O$",O, 1.5*dir(O--relpoint(C--D,2/3)),red); //label(pic1,"$A$",A+(-0.15,0.),blue); label(pic1,"$A$",A,W,blue); label(pic1,"$B$",B, dir(D--B),blue);//dot(B,linewidth(6)+green); //label(pic1,"$C$",C+(0.05,0.2),blue); label(pic1, "$C$",C,E,blue); //label(pic1,"$D$",D+(0.05,-0.2),blue); label(pic1,"$D$",D,dir(B--D),blue); //dot(D,linewidth(6)+red); label(pic1,"$|AC|=1$",D+(0.9,0),blue); perpendicular(pic1, B,SE,B--C,blue); perpendicular(pic1, D,NE,D--C,blue); markangle("$2\alpha$",C,O,B,n=1,radius=7mm,red); markangle("$\alpha$",C,A,B,n=1,radius=7mm,blue); markangle("$\beta$",D,A,C,n=2,radius=7mm,blue); plot_label(pic1, B, C, "$\sin\alpha$",.0875); plot_label(pic1, A, B, "$\cos\alpha$",.0825); plot_label(pic1, D, C, "$\sin\beta$",.1); plot_label(pic1, A, D, "$\cos\beta$",.1); plot_label(pic1, D, B, "$\sin(\alpha+\beta)$",.1); /* dot(pic1,O,blue+3pt); dot(pic1,A,blue+3pt); dot(pic1,B,blue+3pt); dot(pic1,C,blue+3pt);*/ // dot(pic1, O^^A^^B^^C^^D,blue+3pt); real degreesz1z2(pair z1, pair z2) {return degrees(dir(z1 -- z2));} real other_alpha=degreesz1z2(A,B);//55=(90+20)/2 //draw(pic1,arc(A,2*Cos(other_alpha),0,other_alpha));//re-convinces real gr=(sqrt(5)-1)/2; dot(pic1, A^^B^^C^^D,blue+linewidth(gr),UnFill); dot(pic1,O,red,UnFill); add(pic1); shipout(scale(1)*bbox(2mm,Fill(white))); [/asy]$

Figure 6a: $|AC|=1$ Trigonometric Identities in the Circle of Thales

$[asy] import graph; import geometry; import markers; usepackage("amsmath"); //usepackage("mtpro2"); //settings.outformat="pdf"; // output is pdf size(7.4cm, keepAspect=true); defaultpen(fontsize(12pt)); real degreesz1z2(pair z1, pair z2) {return degrees(dir(z1 -- z2));} void plot_label(picture pic, pair A, pair B, string s, real l) { pair AB=B-A; AB=AB/length(AB); real alpha=degrees(acos(AB.x)); if ( AB.y < 0 ) alpha = -alpha; pair normal=(-AB.y,AB.x); label(pic,rotate(alpha)*s,.5(A+B)+l*normal,blue); } picture pic1, pic2, pic3; // pic 2 real alpha=20; real r = 1; pair O=(0,0); real alpha1=20; real alpha2=20+90; real alpha3=180+45; real alpha4=180+75; draw(pic3,unitcircle,red+1.2pt); // pic1 real alpha1=0; real alpha2=180; real alpha3=90+20; real alpha4=180+85; draw(pic1,unitcircle,red+1.2pt); pair A=dir(alpha2); pair C=dir(alpha1); pair B=dir(alpha3); pair D=dir(alpha4); draw(pic1,O--A,black+.8pt); draw(pic1,O--C,black+.8pt); draw(pic1,B--D,black+.8pt); draw(pic1,A--B--C--D--cycle,black+.8pt); draw(pic1, O--B,red); //label(pic1,"$O$",(0.02,-0.2),red); label(pic1,"$O$",O, 1.5*dir(O--relpoint(C--D,2/3)),red); //label(pic1,"$A$",A+(-0.15,0.),blue); label(pic1,"$A$",A,W,blue); label(pic1,"$B$",B, dir(D--B),blue);//dot(B,linewidth(6)+green); //label(pic1,"$C$",C+(0.05,0.2),blue); label(pic1, "$C$",C,E,blue); //label(pic1,"$D$",D+(0.05,-0.2),blue); label(pic1,"$D$",D,dir(B--D),blue); //dot(D,linewidth(6)+red); label(pic1,"$|AC|=2$",D+(0.9,0),blue); perpendicular(pic1, B,SE,B--C,blue); perpendicular(pic1, D,NE,D--C,blue); markangle("$2\alpha$",C,O,B,n=1,radius=7mm,red); markangle("$\alpha$",C,A,B,n=1,radius=7mm,blue); markangle("$\beta$",D,A,C,n=2,radius=7mm,blue); plot_label(pic1, B, C, "$2\sin\alpha$",.0875); plot_label(pic1, A, B, "$2\cos\alpha$",.0825); plot_label(pic1, D, C, "$2\sin\beta$",.1); plot_label(pic1, A, D, "$2\cos\beta$",.1); //plot_label(pic1, D, B, "$2\sin(\alpha+\beta)$",.1); real degreesBD=degreesz1z2(B,D); label(pic1, rotate(degreesBD-180)*"$2\sin(\alpha+\beta)$", relpoint(B--D,.537),.5*dir(W),blue); /* dot(pic1,O,blue+3pt); dot(pic1,A,blue+3pt); dot(pic1,B,blue+3pt); dot(pic1,C,blue+3pt);*/ // dot(pic1, O^^A^^B^^C^^D,blue+3pt); real other_alpha=degreesz1z2(A,B);//55=(90+20)/2 //draw(pic1,arc(A,2*Cos(other_alpha),0,other_alpha));//re-convinces real gr=(sqrt(5)-1)/2; dot(pic1, A^^B^^C^^D,blue+linewidth(gr),UnFill); dot(pic1,O,red,UnFill); add(pic1); shipout(scale(1)*bbox(2mm,Fill(white))); [/asy]$

Figure 6b: $|AC|=2$ Trigonometric Identities in the Circle of Thales

$[asy] import graph; import geometry; real degreesz1z2(pair z1, pair z2) {return degrees(dir(z1 -- z2));} size(8cm); unitsize(.23cm); pair A = (5, 12); pair B = (0, 0); pair C = (5 + 2035/69, 0); pair D = (10, 0); pair myE = (5/2 + 2035/138, -400/23); pair mpBC=midpoint(B--C); pair O=midpoint(B--C); pair Afoot1=(A.x,0); pair Afoot2=foot(A,C,myE); real degreesBA=degreesz1z2(B,A); real degreesBE=degreesz1z2(B,myE); real degreesCE=degreesz1z2(C,myE); real degreesEC=degreesz1z2(myE,C); real degreesEA=degreesz1z2(myE,A); real degreesOA=degreesz1z2(O,A); real theta=degreesz1z2(O,A); fill(A--myE--C--cycle, red+white); //fill(A--B--O--cycle,lightgreen); fill(A--B--D--cycle,mediumred+white); draw(arc(mpBC,2.8,0,degreesz1z2(mpBC,A)),blue); draw(arc(B,3.8,0,degreesBA),blue); //draw(arc(myE,3.8, degreesEC,degreesEA),blue); draw(arc(myE,5.8, 45,45+theta/2),blue); label("$\theta$",mpBC,dir(.5*degreesz1z2(mpBC,A)),blue); label("$\theta/2$",B,dir(.2*degreesz1z2(mpBC,A)),blue); label("$\theta/2$",myE,4.5*dir(myE--C,myE--A),blue); //draw(circle(A,arclength(A--B)));//AB=AD? //draw(circle(A,arclength(A--C)));// AC=AE? //draw(arc(myE,5,degreesEC,degreesEC+90)); draw(A--(A.x,0),red); draw(A--mpBC,red); draw(O--Afoot2,red+Dotted); draw(circumcircle(A,B,C),red); //label("$A$", A, N); label("$A$", A, dir(mpBC--A)); label("$B$", B, W); label("$C$", C, E); label("$D$", D, NE); label("$E$", myE, S); label("$O$",O,dir(O--B,A--O),red); //label("$\alpha$", D, dir(D--B,D--myE));//??? perpfactor=.8; //perpendicularmark(myE,dir(myE--B,myE--C),blue); perpendicularmark(myE,N,blue); perpendicularmark(Afoot1,NE,blue); perpendicularmark(Afoot2,dir(degreesEC+45),blue); perpendicularmark(A,dir(A--B,A--C),blue); draw(A--B--C--cycle); draw(A--D); draw(B--myE--C); draw(D--myE); draw(anglemark(B, A, D, 75)); draw(anglemark(D, A, C, 75)); draw(anglemark(myE, B, D, 75)); pair Ap=(A.x,-A.y); pair F=intersectionpoint(A--myE,Ap--C); pair G=intersectionpoint(B--myE,Ap--C); filldraw(B--Ap--C--cycle,green); label("$A^\prime$",Ap,dir(O--Ap)); label("$F$",F,1.25*dir(A--myE,C--Ap)*dir(-3)); label("$G$",G,1.0*dir(B--myE,C--Ap)*dir(-8)); draw(B--myE^^D--myE); perpendicularmark(F,dir(F--A,F--C),blue); perpendicularmark(Ap,dir(Ap--B,Ap--G),blue); dot(A^^Ap^^B^^C^^D^^myE^^F^^G^^O,red,Fill(white)); shipout(bbox(2mm,Fill(white))); [/asy]$

Figure 7: Quadrilateral (Four-sided Figure) $ABA\rq C$
A Kite - composed of two Halves, which are same Right Triangles