diff --git a/frontend/textbooks/geometry_8_ch4.html b/frontend/textbooks/geometry_8_ch4.html
index 6ed40bf..8a3847c 100644
--- a/frontend/textbooks/geometry_8_ch4.html
+++ b/frontend/textbooks/geometry_8_ch4.html
@@ -2120,27 +2120,42 @@ function buildP5(){
     $$|BT_1| = |BT_2|, \\quad |BT_3| = |BT_4|$$
     <p style="margin-top:8px">Формула радиуса: $r = d\\cdot\\sin\\dfrac{\\alpha}{2}$, где $d = |BO|$ — расстояние от вершины до центра, $\\alpha$ — угол.</p>
     <div style="display:flex;justify-content:center;margin-top:10px">
-      <svg viewBox="0 0 275 195" style="max-width:285px;background:#fafafa;border:1px solid var(--border);border-radius:10px">
-        <!-- B=(15,178), half-angle=15°, side1=horizontal, side2=30° above horiz -->
-        <!-- bisector dir=(cos15°,-sin15°)=(0.966,-0.259) -->
-        <!-- O₁: r=22, d=84.9 → O₁=(97,156); O₂: r=36, d=139 → O₂=(149,142) -->
+      <svg viewBox="0 0 280 200" style="max-width:300px;background:#fafafa;border:1px solid var(--border);border-radius:10px">
+        <!-- B=(15,178), full angle = 30°, half = 15°.
+             Lower side: horizontal y=178, to (265,178).
+             Upper side: direction (cos30°,-sin30°)=(0.866,-0.5). At length 280 → (15+242.5, 178-140)=(257,38).
+             Bisector: (cos15°,-sin15°)=(0.966,-0.259). To length 220 → (15+212.5, 178-57)=(227,121).
+             O₁: d=84.9, O₁=(15+82, 178-22)=(97,156), r=d·sin15°=22.
+             O₂: d=139, O₂=(15+134, 178-36)=(149,142), r=d·sin15°=36.
+             Upper tangent points: foot of perpendicular from O onto upper line.
+             T₁_up = B + ((O₁-B)·u) · u, where u=(0.866,-0.5).
+               (O₁-B)·u = 82·0.866 + (-22)·(-0.5) = 71+11 = 82.
+               T₁_up = (15+82·0.866, 178+82·(-0.5)) = (86,137).
+             T₂_up = (15+134·0.866, 178+134·(-0.5)) = (131,111). -->
         <line x1="15" y1="178" x2="265" y2="178" stroke="#0891b2" stroke-width="2"/>
-        <line x1="15" y1="178" x2="258" y2="56" stroke="#0891b2" stroke-width="2"/>
-        <line x1="15" y1="178" x2="220" y2="127" stroke="#64748b" stroke-width="1.3" stroke-dasharray="4,3"/>
+        <line x1="15" y1="178" x2="257" y2="38" stroke="#0891b2" stroke-width="2"/>
+        <line x1="15" y1="178" x2="227" y2="121" stroke="#64748b" stroke-width="1.3" stroke-dasharray="4,3"/>
         <!-- small circle O₁=(97,156) r=22 -->
         <circle cx="97" cy="156" r="22" fill="rgba(8,145,178,.09)" stroke="#0891b2" stroke-width="1.8"/>
         <circle cx="97" cy="156" r="2.5" fill="#0e7490"/>
         <!-- large circle O₂=(149,142) r=36 -->
         <circle cx="149" cy="142" r="36" fill="rgba(8,145,178,.05)" stroke="#0891b2" stroke-width="1.8"/>
         <circle cx="149" cy="142" r="2.5" fill="#0e7490"/>
-        <!-- tangent points: T₁_h=(97,178), T₁_s=(86,137), T₂_h=(149,178), T₂_s=(131,111) -->
+        <!-- tangent points on lower side -->
         <circle cx="97" cy="178" r="3" fill="#10b981"/>
         <circle cx="149" cy="178" r="3" fill="#10b981"/>
-        <circle cx="15" cy="178" r="3" fill="#0e7490"/>
-        <text x="3" y="175" font-size="10" font-weight="700" fill="#0e7490">B</text>
-        <!-- O labels just outside circle boundary to avoid overlap -->
-        <text x="100" y="154" font-size="9" fill="#0e7490" font-weight="700">O₁</text>
-        <text x="152" y="140" font-size="9" fill="#0e7490" font-weight="700">O₂</text>
+        <!-- tangent points on upper side -->
+        <circle cx="86" cy="137" r="3" fill="#10b981"/>
+        <circle cx="131" cy="111" r="3" fill="#10b981"/>
+        <!-- B point -->
+        <circle cx="15" cy="178" r="3.5" fill="#0e7490"/>
+        <text x="3" y="175" font-size="11" font-weight="700" fill="#0e7490">B</text>
+        <text x="89" y="190" font-size="9" fill="#065f46" font-weight="700">T₁</text>
+        <text x="141" y="190" font-size="9" fill="#065f46" font-weight="700">T₂</text>
+        <text x="63" y="135" font-size="9" fill="#065f46" font-weight="700">T₃</text>
+        <text x="108" y="110" font-size="9" fill="#065f46" font-weight="700">T₄</text>
+        <text x="91" y="160" font-size="10" fill="#0e7490" font-weight="700">O₁</text>
+        <text x="143" y="146" font-size="10" fill="#0e7490" font-weight="700">O₂</text>
       </svg>
     </div>`);