diff --git a/frontend/textbooks/geometry_8_ch1.html b/frontend/textbooks/geometry_8_ch1.html
index 9f5d2a6..3a0f8f9 100644
--- a/frontend/textbooks/geometry_8_ch1.html
+++ b/frontend/textbooks/geometry_8_ch1.html
@@ -1802,7 +1802,7 @@ function buildP4(){
     const W=300,H=200;
     function drawProof(highlight){
       const A={x:30,y:160},B={x:180,y:160},C={x:260,y:50},D={x:110,y:50};
-      const cx=(A.x+B.x+C.x+D.x)/4;
+      const cx=(A.x+B.x+C.x+D.x)/4, cy=(A.y+B.y+C.y+D.y)/4;
       let s='<svg viewBox="0 0 '+W+' '+H+'" style="width:100%;max-width:320px;background:var(--card);border:1px solid var(--border);border-radius:12px">';
       const t1fill=highlight==='diag'||highlight==='both'?'rgba(220,38,38,.15)':'rgba(220,38,38,.07)';
       const t2fill=highlight==='both'?'rgba(16,185,129,.15)':'rgba(220,38,38,.07)';